How statement and proof provenance work

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Electrostatic boundaries and conductors

1 · Prerequisites

2 · Summary

This page develops the ideal electrostatic conductor model and the boundary value problems it induces in one named inertial laboratory frame. The ideal electrostatic conductor model adopts the equilibrium idealisation: finitely many compact smooth field-free conductor bodies strictly inside an ambient region, a connected vacuum region, a continuous stationary surface charge, the distributional vacuum Maxwell equations, and a continuous equilibrium potential that is constant on each conductor body, with the two data types of prescribed component potentials or prescribed component charges. Three local mathematical prerequisites supply the geometry and the uniqueness tool: The boundary of a bounded smooth domain is a compact embedded hypersurface converts the one-sided graph definition of a bounded smooth domain into a compact embedded hypersurface with finitely many boundary components; The vacuum region around compact smooth conductors is a bounded smooth domain shows that the complement of the conductor bodies is again a bounded smooth domain with the expected outer and conductor boundary pieces and normal signs; and Uniqueness for harmonic functions on a punctured half-space with zero trace, decaying at the pole and at infinity gives the decay-class uniqueness used for the unbounded image problem. Surface charge and normal field on an ideal conductor proves the pointwise surface-charge and normal-field relations, the constancy of the potential trace on each conductor component, and the total-charge flux identities. Uniqueness and flux compatibility for the electrostatic boundary value problem and Existence and regularity for the electrostatic boundary value problem establish the classical, weak and transmission uniqueness, flux compatibility and smooth solvability statements on the bounded vacuum region. The image-method certificate for a point charge above a grounded plane verifies the classic grounded-plane image solution as an explicitly declared idealisation of the half-space, and Capacitance and energy for specified linear conductor geometry derives the linear charge-potential map, the symmetric positive definite capacitance matrix and the stored-energy identities for the specified conductor geometry.

The companion page em-electrostatic-boundaries-and-conductors-examples carries the worked grounded-plane force, the ideal parallel-plate capacitor and the incompatible-Neumann-data impossibility example.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

The boundary of a bounded smooth domain is a compact embedded hypersurface

Statement

Assume the Axiom of Countable Choice ( ${A C}_{ω}$ ), the convention of the boundary-chart and surface-measure interfaces cited below (The Axiom of Countable Choice ( ${A C}_{ω}$ )); no other choice principle is used. Let $k \geq 1$ and let $D \subset R^{3}$ be a bounded $C^{k}$ domain in the sense of Bounded C^k domains and boundary charts (nonempty, bounded, open, and locally exactly the one-sided subgraph of a $C^{k}$ graph function over a hyperplane). Then:

(a) [Compactness and finiteness] $\partial D$ is nonempty and compact; the complement $R^{3} ∖ \partial D$ need not be connected here. Moreover $\partial D$ has finitely many connected components, and each connected component of $\partial D$ is compact.

(b) [Finite embedded presentation] the local boundary charts give finitely many regular injective $C^{k}$ parametrisations of $\partial D$ with $C^{k}$ coordinate transitions, of the graph type required by Surface integration on compact C1 hypersurfaces: writing a chart as $R (p) = (y, s)$ , the graph map $y \mapsto R^{- 1} (y, h (y))$ is injective, of class $C^{k}$ , with $C^{k}$ inverse (the inverse is $p \mapsto π (R (p))$ restricted to the graph, where $π (y, s) = y$ ); consequently $\partial D$ is a compact embedded $C^{k}$ hypersurface, and for $k \geq 1$ it carries the finite surface measure of Surface integration on compact C1 hypersurfaces and Chart and partition independence of surface measure.

(c) [Normal field] the outward unit normal $ν$ of Bounded C1 domains and their outward normals is a continuous unit normal field on $\partial D$ , with $ν (x)$ orthogonal to the tangent plane of the graph chart at each $x$ ; it is the field used by the Green and divergence items cited by this page's consumers.

The statement is geometric: it asserts no differentiability of $D$ beyond $C^{k}$ , no boundedness of chart derivatives beyond the compact patches, and no connectedness of either $D$ or $\partial D$ .

Facts & Assumptions

Given: Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $k \geq 1$ and let $D \subset R^{3}$ be a bounded $C^{k}$ domain; write $\partial D$ for its metric boundary (Interior, closure, boundary, limit point, isolated point and dense subset of a metric space) and use the Euclidean metric $d_{2}$ with induced norm $∥ \cdot ∥_{2}$ on $R^{3}$ ( $R^{n}$ as the set of functions $n \to R$ , and $d_{1}$ , $d_{2}$ , $d_{\infty}$ are metrics on it, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation).

[F1]

The chart property: for every $x \in \partial D$ there are an open $W \subseteq R^{3}$ with $x \in W$ , a rigid motion $R (p) = Q p + b$ with orthogonal $Q$ , an open ball $B \subseteq R^{2}$ and $h \in C^{k} (B; R)$ with $R (D \cap W) = R (W) \cap {(y, s) : s < h (y)}$ ; in particular $\partial D \cap W = R^{- 1} ({(y, s) : s = h (y)} \cap R (W))$ (Bounded C^k domains and boundary charts). Shrinking $W$ preserves this identity.

[F2]

In such coordinates the outward normal is $ν = (- D h (y), 1) / \sqrt{1 + ∣ D h (y) ∣^{2}}$ transported back to ambient coordinates by the inverse orthogonal map $Q^{T}$ , and it agrees on chart overlaps (Bounded C1 domains and their outward normals, Chart and partition independence of surface measure).

[F3]

Surface integration on a compact embedded $C^{1}$ hypersurface is built from finitely many regular injective $C^{1}$ parametrisations with $C^{1}$ transitions and an ambient partition subordinate to the chart neighbourhoods (Surface integration on compact C1 hypersurfaces).

[F4]

For any compact Euclidean set covered by finitely many open sets $U_{j}$ there are smooth nonnegative $χ_{j}$ with compact support in $U_{j}$ and $\sum_{j} χ_{j} = 1$ on a neighbourhood of the set (Finite ambient partitions near compact sets).

[F5]

The chart integral of [F3] defines a finite Borel measure independent of charts and partitions; in graph coordinates its density is $\sqrt{1 + ∣ D h ∣^{2}}$ ; on a one-sided domain boundary the outward unit normal of [F2] is continuous (Chart and partition independence of surface measure).

[F6]

Metric boundary, interior and closure: $\partial A = \overline{A} ∖ int (A)$ , $int (A) \subseteq A \subseteq \overline{A}$ , every ball about an adherent point meets $A$ , and $int (A)$ is open (Interior, closure, boundary, limit point, isolated point and dense subset of a metric space).

[F7]

A set is closed exactly when its complement is open (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement). Arbitrary unions and finite intersections of open sets are open (Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed); by taking complements and applying De Morgan's laws, arbitrary intersections and finite unions of closed sets are closed. Complements interchange open and closed sets.

[F8]

$\overline{A}$ is closed, contains $A$ , and is contained in every closed superset of $A$ ; $A$ is closed if and only if $A = \overline{A}$ (The closure of a nonempty $A$ is ${x : d (x, A) = 0}$ , equals $A$ together with its limit points, and is the smallest closed superset).

[F9]

A subset of $R^{n}$ is compact exactly when it is closed and bounded (Open cover, subcover, compact metric space, and compact subset of a metric space, Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line). For a compact subset, an ambient open cover has a finite subcover by A subset of a metric space is open in the subspace metric exactly when it is the trace of an open set of the ambient space, and it is compact as a metric space in its own right exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it.

[F10]

$A$ is bounded when $A \subseteq B (x_{0}, r)$ for some $x_{0}$ and $r > 0$ , where $B (x_{0}, r) = {y : d_{2} (y, x_{0}) < r}$ ; distances satisfy the triangle inequality (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space, Open ball, closed ball and sphere in a metric space, Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric).

[F11]

$R^{3}$ is polygonally connected and connected, and every path-connected space is connected ( $R^{n}$ is polygonally connected, connected, locally path-connected and locally connected, Every path-connected space is connected, and every path component lies inside a component).

[F12]

A continuous image of a connected space is connected (A continuous image of a connected space is connected, and connectedness is a topological property).

[F13]

The only clopen subsets of a connected space are the empty set and the whole space (For a topological space the following agree: no separation exists, the only clopen subsets are $\emptyset$ and $X$ , and every continuous map to the two-point discrete space is constant).

[F14]

Components are nonempty, pairwise disjoint, cover the space, are maximal connected sets, and are closed (Connected components, quasicomponents, and totally disconnected spaces, The components of a space are its maximal connected subsets, they partition it, and each of them is closed).

[F15]

A union of connected sets with a common point is connected (A union of connected subspaces with a point in common is connected, and so is a union of a family in which every member meets a fixed connected member).

[F16]

Local connectedness at $x$ means that every open neighbourhood of $x$ contains an open connected neighbourhood of $x$ ; a set that is locally connected at each of its points is locally connected (Locally connected and locally path-connected spaces: a neighbourhood base of open connected, respectively open path-connected, sets at every point).

[F17]

$C^{k}$ maps between open Euclidean sets, their composition rule, the chain rule, and the algebra of derivatives; $C^{k}$ means all partial derivatives through order $k$ exist and are continuous, and total differentiability implies continuity ( $C^{k}$ Euclidean maps and diffeomorphisms, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Total differentiability gives a local $O (∥ h ∥_{2})$ increment bound and therefore continuity).

[F18]

The Euclidean norm is a norm inducing $d_{2}$ , so it is homogeneous and satisfies the triangle inequality; orthogonal maps are invertible linear isometries and preserve norms and distances, hence are continuous in both directions (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation, A norm on a real vector space, the induced metric, and the dictionary with the metric axioms, Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces, Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form).

[F19]

Every norm on $R^{n}$ is continuous for $d_{2}$ , and here the coordinate basis is labelled from one: $e_{j} = e_{j - 1}^{c a n}$ for $j = 1, 2, 3$ . Thus $e_{1}$ denotes the first coordinate vector, with $∥ e_{1} ∥_{2} = 1$ (The finite and reverse triangle inequalities for a norm; and for $n \geq 1$ every norm $N$ on $R^{n}$ satisfies $N (x) \leq C ∥ x ∥_{1}$ and is Lipschitz, hence continuous, for $d_{2}$ , The standard list $e : n \to F^{n}$ with $e_{i} (i) = 1_{F}$ and $e_{i} (j) = 0_{F}$ for $j \neq i$ is an ordered basis of $F^{n}$ ; hence $\dim_{F} F^{n} = n$ , and $F^{0}$ is the zero space with basis $\emptyset$ and dimension $0$ ).

[F20]

The open ball $B_{R} (a) \subseteq R^{n}$ , $n \geq 2$ , is a bounded $C^{1}$ domain with boundary the Euclidean sphere $S_{R} (a)$ ; the local graph function $h (w) = \sqrt{R^{2} - ∣ w ∣^{2}} - R$ is $C^{\infty}$ on $∣ w ∣ < R$ , by iterated differentiation of the real power $s \mapsto s^{1 / 2}$ on $(0, \infty)$ (Continuity and derivatives of positive-base real powers) composed with the polynomial $R^{2} - ∣ w ∣^{2}$ (Euclidean balls are bounded C-one domains with radial outward normal, Euclidean spheres and closed balls as subspaces of $R^{n}$ , $C^{k}$ Euclidean maps and diffeomorphisms).

Proof

technique · direct

1.1F6F7F8

$\partial D$ is closed. By [F6], $\partial D = \overline{D} ∖ int (D) = \overline{D} \cap (R^{3} ∖ int (D))$ , an intersection of the closed set $\overline{D}$ [F8] with the complement of the open set $int (D)$ [F6], hence closed [F7].

1.2F6F10

$\partial D$ is bounded. Since $D$ is bounded, fix $x_{0} \in R^{3}$ and $r > 0$ with $D \subseteq B (x_{0}, r)$ [F10]. Let $x \in \overline{D}$ . By [F6] the ball $B (x, 1)$ meets $D$ , so there is $y \in D$ with $d_{2} (x, y) < 1$ ; then $d_{2} (y, x_{0}) < r$ and the triangle inequality gives $d_{2} (x, x_{0}) \leq d_{2} (x, y) + d_{2} (y, x_{0}) < 1 + r$ [F10]. Hence $\overline{D} \subseteq B (x_{0}, 1 + r)$ is bounded, and $\partial D \subseteq \overline{D}$ is bounded [F10, F6].

1.3F1F17given

Chart patches. Let $x \in \partial D$ and let $G$ be an open neighbourhood of $x$ in $R^{3}$ . Apply [F1] to get $W$ , $R$ , $B$ , $h$ with the chart identity; replacing $W$ by the open set $W \cap G$ preserves it [F1], so assume $W \subseteq G$ . Write $R (p) = (y, s)$ , $π (y, s) = y$ , put $y_{x} : = π (R (x))$ , and set $B^{''} : = π (R (W) \cap {(y, s) : s = h (y)})$ . This is an open subset of $B$ containing $y_{x}$ : the set $R (W) \cap {s = h (y)}$ is relatively open in the graph $Γ = {(y, h (y)) : y \in B}$ , and $π ∣_{Γ} : Γ \to B$ is a homeomorphism: its inverse $y \mapsto (y, h (y))$ and the projection are continuous by [F17], so its image is open; and $y_{x} \in B^{''}$ because $x \in \partial D \cap W$ maps to a point of the graph [F1]. Define $X : B^{''} \to R^{3}$ , $X (y) : = R^{- 1} (y, h (y))$ . Then $X (B^{''}) = \partial D \cap W$ : every $X (y)$ lies in $\partial D \cap W$ by [F1], and conversely for $p \in \partial D \cap W$ the point $R (p) \in R (W)$ has $s = h (y)$ [F1], so $π (R (p)) \in B^{''}$ and $X (π (R (p))) = R^{- 1} (y, h (y)) = R^{- 1} (R (p)) = p$ . Moreover $X$ is injective with inverse $π \circ R$ restricted to $X (B^{''})$ , because $π (R (X (y))) = y$ and $X (π (R (p))) = p$ for $p = X (y)$ .

2.1F6F8F10F11F13F18F19step 1.2

$\partial D \neq \emptyset$ . Suppose $\partial D = \emptyset$ . Then $\overline{D} \subseteq int (D)$ by [F6], and since $int (D) \subseteq D \subseteq \overline{D}$ [F6] we get $D = \overline{D}$ , so $D$ is closed [F8] as well as open; that is, $D$ is clopen, and $D \neq \emptyset$ by hypothesis. Moreover $D \neq R^{3}$ : if $D = R^{3}$ then $R^{3} = D \subseteq B (x_{0}, r)$ for the ball of step 1.2, but the point $w : = x_{0} + (r + 1) e_{1}$ satisfies $d_{2} (w, x_{0}) = ∥ (r + 1) e_{1} ∥_{2} = (r + 1) ∥ e_{1} ∥_{2} = r + 1 > r$ , so $w \in R^{3} ∖ B (x_{0}, r)$ [F18, F19]. Thus $D$ is a proper nonempty clopen subset of the connected space $R^{3}$ [F11], contradicting [F13]. Hence $\partial D \neq \emptyset$ .

2.2F9step 1.1step 1.2

$\partial D$ is compact by [F9], being closed [step 1.1] and bounded [step 1.2].

2.3F17F18step 1.3

$X$ is of class $C^{k}$ , regular, and its inverse is $C^{k}$ . The map $g (y) : = (y, h (y))$ has components the coordinate functions and $h$ , all $C^{k}$ , so $g$ is $C^{k}$ with all partial derivatives continuous [F17]; $R^{- 1} (z) = Q^{T} (z - b)$ is affine with orthogonal linear part, hence $C^{\infty}$ , continuous, with continuous inverse [F18]. By the chain rule $X = R^{- 1} \circ g$ is $C^{k}$ , hence continuous, with $D X (y) = Q^{T} \circ D g (y)$ [F17]. The differential $D g (y)$ has first two rows the identity, hence is injective, and $Q^{T}$ is invertible, so $D X (y)$ is injective at every $y$ ; thus $X$ is a regular injective parametrisation, regularity being exactly the injectivity of $D X (y)$ just proved. The inverse $π \circ R ∣_{X (B^{''})}$ is the restriction of the continuous map $π \circ R$ [F18] and is continuous at every point of $X (B^{''})$ ; so $X$ is a homeomorphism onto $X (B^{''})$ .

3.1F7F11F12F16F18step 1.3step 2.3

Connected sub-patches. Let $B_{1}$ be an open ball with $y_{x} \in B_{1}$ and $\overline{B_{1}} \subseteq B^{''}$ , and put $O_{1} : = W \cap R^{- 1} (B_{1} \times R)$ , an open neighbourhood of $x$ . Then $X (B_{1}) = \partial D \cap O_{1}$ : the inclusion $\subseteq$ is [step 1.3] with $y \in B_{1} \subseteq B^{''}$ ; conversely $p \in \partial D \cap O_{1}$ has $π (R (p)) \in B^{''} \cap B_{1} = B_{1}$ [step 1.3]; this set is relatively open in $\partial D$ because $O_{1}$ is open in $R^{3}$ [F7]. The ball $B_{1}$ is path-connected: for $u, v \in B_{1}$ and $t \in [0, 1]$ the triangle inequality and homogeneity of the norm give $∥ (1 - t) u + t v - y_{x} ∥_{2} \leq (1 - t) ∥ u - y_{x} ∥_{2} + t ∥ v - y_{x} ∥_{2} < (1 - t) ρ + t ρ = ρ$ when $B_{1} = B (y_{x}, ρ)$ , so the segment $t \mapsto (1 - t) u + t v$ stays in $B_{1}$ [F18]; thus $B_{1}$ is path-connected, hence connected [F11]. As the continuous image of $B_{1}$ under $X$ [step 2.3] the set $X (B_{1})$ is connected [F12]. Since $B_{1}$ can be chosen with arbitrarily small radius, $\partial D$ is locally connected [F16].

3.2F2F5F17F19step 1.3step 2.3

The normal field. In the coordinates of [step 1.3] the vector $(- D h_{j} (y), 1)$ is orthogonal to every tangent column $(e_{i}, \partial_{i} h_{j} (y))$ of [step 2.3] because $- (\partial_{i} h_{j}) (y) + (\partial_{i} h_{j}) (y) = 0$ , and it is nonzero, so $(- D h_{j}, 1) / \sqrt{1 + ∣ D h_{j} ∣^{2}}$ is a unit normal field of the graph; by [F2] it is the prescribed outward normal $ν$ , transported back by $Q_{j}^{T}$ . Since $h_{j}$ is $C^{k}$ with $k \geq 1$ , $D h_{j}$ is continuous, and the norm of $(- D h_{j} (y), 1)$ is $\sqrt{1 + ∣ D h_{j} (y) ∣^{2}} \geq 1$ , so $y \mapsto ν (X_{j} (y))$ is continuous as a composition of continuous maps on each patch [F17, F19]; by [F5] the field $ν$ is defined consistently on overlaps and is continuous on all of $\partial D$ . This proves (c).

4.1F7F14F15step 3.1

Components are open. Let $C$ be a connected component of $\partial D$ [F14] and let $x \in C$ . By [step 3.1] and [F16] applied to the open set $\partial D$ itself, there is a connected open-in- $\partial D$ set $V$ with $x \in V$ . Then $V \cup C$ is connected [F15], contains $x$ , and contains the maximal connected set $C$ [F14], so $V \cup C = C$ , i.e. $V \subseteq C$ . Hence every point of $C$ has a neighbourhood contained in $C$ , so $C$ is open in $\partial D$ [F7].

4.2F3F4F5F17step 2.2step 1.3step 2.3step 3.1

Finite embedded presentation. The family of all open balls $B_{1}$ with $\overline{B_{1}} \subseteq B^{''}$ for charts of [step 1.3] over all $x \in \partial D$ produces open sets $O_{1} = W \cap R^{- 1} (B_{1} \times R)$ whose patches $X (B_{1}) = \partial D \cap O_{1}$ [step 3.1] cover $\partial D$ ; compactness [step 2.2] gives a finite subcover $O_{1}, \dots, O_{N}$ with parametrisations $X_{j} : B_{1, j} \to \partial D \cap O_{j}$ , each regular, injective and $C^{k}$ [step 2.3] with connected open parameter balls. Let $T : = X_{ℓ}^{- 1} \circ X_{j}$ be a transition, defined on the open set $X_{j}^{- 1} (\partial D \cap O_{j} \cap O_{ℓ}) \subseteq B_{1, j}$ : it is a composition of the homeomorphisms of [step 1.3] and [step 2.3], hence a bijection onto its image, and it is given by $T (y) = π (R_{ℓ} (R_{j}^{- 1} (y, h_{j} (y))))$ , a composition of affine $C^{\infty}$ maps with the $C^{k}$ map $g_{j} (y) = (y, h_{j} (y))$ and the linear projection $π$ ; so $T$ is $C^{k}$ with $C^{k}$ inverse $X_{j}^{- 1} \circ X_{ℓ}$ [F17]. With $K : = \partial D$ and the finitely many open sets $O_{j}$ covering it, [F4] supplies smooth $χ_{j}$ with compact support in $O_{j}$ and $\sum_{j} χ_{j} = 1$ on a neighbourhood of $\partial D$ . This is exactly the finite presentation used by the surface integral [F3]; by [F5] the chart integral then defines the finite Borel surface measure of $\partial D$ , independent of the charts and partition, with graph density $\sqrt{1 + ∣ D h_{j} ∣^{2}}$ , and $\partial D$ is a compact embedded $C^{k}$ hypersurface. This proves (b).

5.1F9F14step 2.2step 4.1

Finitely many components. The components of $\partial D$ are nonempty, pairwise disjoint, cover $\partial D$ [F14], and are open [step 4.1]; being pairwise disjoint, every subfamily with the same union has the same members. Hence they form an open cover of the compact space $\partial D$ [step 2.2], which therefore has a finite subcover [F9]; any component meets that finite union and is therefore one of its finitely many members. So $\partial D$ has finitely many components; each is closed [F14] and bounded (being a subset of bounded $\partial D$ ), hence compact [F9]. This proves (a).

6.1F1F6F7F20∎

The complement need not be connected. Let $c_{1} : = 0$ , $c_{2} : = 4 e_{1}$ and $D_{0} : = B_{1} (c_{1}) \cup B_{1} (c_{2})$ . Each ball is a bounded $C^{1}$ domain whose local graph function is $C^{\infty}$ [F20]; since $d_{2} (c_{1}, c_{2}) = 4 > 2$ , the closed balls are disjoint, and at a boundary point of $B_{1} (c_{1})$ the chart of [F20] shrunk to the open complement of the closed ball around $c_{2}$ exhibits $D_{0}$ locally as the same subgraph, and symmetrically, so $D_{0}$ is a bounded $C^{k}$ domain for every $k \geq 1$ [F1, F20]. Its boundary is $\partial D_{0} = S_{1} (c_{1}) \cup S_{1} (c_{2})$ : a point of either sphere lies in $\overline{D_{0}}$ but, every ball about it meeting the exterior of that ball, outside $int (D_{0})$ [F6, F20], while a point of $B_{1} (c_{1})$ or $B_{1} (c_{2})$ lies in the interior of $D_{0}$ , and a point outside both closed balls has a ball avoiding $D_{0}$ entirely [F6, F7, F20]. Now put $U : = B_{1} (c_{1})$ and $V : = B_{1} (c_{2}) \cup (R^{3} ∖ ({\overline{B}}_{1} (c_{1}) \cup {\overline{B}}_{1} (c_{2})))$ . Both are open [F7, F20], nonempty, and disjoint, because $B_{1} (c_{2}) \subseteq {\overline{B}}_{1} (c_{2})$ is disjoint from the exterior term; and $U \cup V = R^{3} ∖ \partial D_{0}$ , since the removed set $\partial D_{0}$ is exactly the union of the two spheres. Thus $U$ and $V$ , open in the subspace $R^{3} ∖ \partial D_{0}$ , nonempty, disjoint and covering it, form a separation of the complement [F6, F7]. This witnesses the asserted non-claim for every $k \geq 1$ .

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

The vacuum region around compact smooth conductors is a bounded smooth domain

Statement

Assume the Axiom of Countable Choice ( ${A C}_{ω}$ ) as carried by the boundary interfaces (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $k \geq 1$ , let $Ω \subset R^{3}$ be a bounded $C^{k}$ domain, and let $D_{1}, \dots, D_{m}$ ( $m \geq 0$ ) be bounded $C^{k}$ domains with pairwise disjoint closures whose closures satisfy $\overline{D_{1}} \cup \dots \cup \overline{D_{m}} \subset Ω$ strictly inside $Ω$ . Put

$Ω_{v} : = Ω ∖ (\overline{D_{1}} \cup \dots \cup \overline{D_{m}}) .$

Then:

(a) [Domain] $Ω_{v}$ is a bounded $C^{k}$ domain in the sense of Bounded C^k domains and boundary charts (in particular nonempty, bounded and open);

(b) [Boundary] $\partial Ω_{v} = \partial Ω \cup \partial D_{1} \cup \dots \cup \partial D_{m}$ , a disjoint union of nonempty compact sets; the compact embedded $C^{k}$ hypersurfaces $\partial Ω, \partial D_{1}, \dots, \partial D_{m}$ are pairwise disjoint (The boundary of a bounded smooth domain is a compact embedded hypersurface);

(c) [Normals and sign] the outward unit normal of $Ω_{v}$ at a point of $\partial Ω$ equals the outward unit normal of $Ω$ , and at a point of $\partial D_{j}$ it equals the negative of the outward unit normal of $D_{j}$ defined by Bounded C^k domains and boundary charts and Bounded C1 domains and their outward normals; so the outward normal of $Ω_{v}$ on $\partial D_{j}$ points from the vacuum into the conductor;

(d) [Componentwise statement] $Ω_{v}$ has finitely many connected components. Each component $U$ is an open bounded $C^{k}$ domain; its boundary is the compact restricted subset $\partial U \subseteq \partial Ω_{v}$ , a union of the boundary components incident to $U$ . The normal formulas in (c) restrict to those boundary pieces. It need not have every boundary piece of $Ω_{v}$ , and connectedness of $Ω_{v}$ is not asserted.

No regularity beyond $C^{k}$ , no convexity and no topological simplicity is claimed.

Facts & Assumptions

Given: ${A C}_{ω}$ as in The Axiom of Countable Choice ( ${A C}_{ω}$ ); a bounded $C^{k}$ domain $Ω \subset R^{3}$ , $k \geq 1$ ; bounded $C^{k}$ domains $D_{1}, \dots, D_{m}$ ( $m \geq 0$ ) with pairwise disjoint closures all strictly contained in $Ω$ . All metric notions use the Euclidean metric $d_{2}$ and norm on $R^{3}$ ; write $B (x, r)$ for open balls, $\overline{A}$ for closure and $\partial A$ for the metric boundary.

[F1]

A bounded set lies in some ball; closures of bounded sets are bounded; a subset of $R^{n}$ is compact exactly when it is closed and bounded; a finite union of compact or of closed sets is closed, and a finite union of bounded sets is bounded (Open ball, closed ball and sphere in a metric space, Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space, Open cover, subcover, compact metric space, and compact subset of a metric space, Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement). Ambient open covers of compact subsets have finite subcovers (A subset of a metric space is open in the subspace metric exactly when it is the trace of an open set of the ambient space, and it is compact as a metric space in its own right exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it).

[F2]

Closure is the smallest closed superset: $A \subseteq F$ with $F$ closed implies $\overline{A} \subseteq F$ , and $A$ is closed iff $A = \overline{A}$ . The boundary is $\partial A = \overline{A} ∖ int (A)$ , and for open $A$ the interior is $A$ itself; boundaries are closed; every ball about a point of $\overline{A}$ meets $A$ (The closure of a nonempty $A$ is ${x : d (x, A) = 0}$ , equals $A$ together with its limit points, and is the smallest closed superset, Interior, closure, boundary, limit point, isolated point and dense subset of a metric space).

[F3]

Distances: $d (x, A)$ is the infimum of ${d (x, a) : a \in A}$ for nonempty $A$ ; a continuous real function on a nonempty compact metric space attains its minimum; $∣ d (x, a) - d (x, a^{'}) ∣ \leq d (a, a^{'})$ , so $q \mapsto d (x, q)$ is continuous, and if $K$ is compact with $x \notin K$ then $d (x, K) > 0$ (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space, A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value, Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric, Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form, The finite and reverse triangle inequalities for a norm; and for $n \geq 1$ every norm $N$ on $R^{n}$ satisfies $N (x) \leq C ∥ x ∥_{1}$ and is Lipschitz, hence continuous, for $d_{2}$ ).

[F4]

The chart property and the outward normal: at every boundary point $x$ of a bounded $C^{k}$ domain $D$ there are an open $W ∋ x$ , a rigid motion $R = Q \cdot + b$ with orthogonal $Q$ , an open ball $B \subseteq R^{2}$ , $h \in C^{k} (B)$ with $R (D \cap W) = R (W) \cap {(y, s) : s < h (y)}$ , and the prescribed outward normal in ambient coordinates is $Q^{T} (- D h (y), 1) / \sqrt{1 + ∣ D h (y) ∣^{2}}$ , agreeing on overlaps; shrinking $W$ preserves the identity (Bounded C^k domains and boundary charts, Bounded C1 domains and their outward normals, The boundary of a bounded smooth domain is a compact embedded hypersurface).

[F5]

For a bounded $C^{k}$ domain the boundary is nonempty, compact, and a compact embedded $C^{k}$ hypersurface with finitely many connected components; hence also $\partial D \cap W = R^{- 1} ({(y, h (y)) : y \in B} \cap R (W))$ and points of $R (W)$ with $s > h (y)$ lie outside $\overline{D}$ (The boundary of a bounded smooth domain is a compact embedded hypersurface).

[F6]

Let $B_{1} \subseteq R^{2}$ be a ball, $h \in C (B_{1}; R)$ , $c \in R$ and $δ > 0$ , and suppose $∣ h (y) - c ∣ \leq δ / 4$ for all $y \in B_{1}$ . With $H : = c + δ / 2$ one then has $h (y) < H < h (y) + δ$ for every $y \in B_{1}$ . Consequently each of the regions $S_{+} : = {(y, s) : y \in B_{1}, h (y) < s < h (y) + δ}$ and $S_{-} : = {(y, s) : y \in B_{1}, h (y) - δ < s < h (y)}$ is polygonally connected: for $u = (y_{1}, s_{1})$ and $v = (y_{2}, s_{2})$ in $S_{+}$ the segment $[y_{1}, y_{2}]$ lies in the convex ball $B_{1}$ (Open ball, closed ball and sphere in a metric space, A norm on a real vector space, the induced metric, and the dictionary with the metric axioms, Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation, $R^{n}$ as the set of functions $n \to R$ , and $d_{1}$ , $d_{2}$ , $d_{\infty}$ are metrics on it), every point $(y, H)$ of the middle segment satisfies $h (y) < H < h (y) + δ$ by the displayed inequality, and the vertical segments from $s_{1}$ up to $H$ at $y_{1}$ and from $H$ down to $s_{2}$ at $y_{2}$ have all their intermediate heights between the endpoint heights, hence also lie in $S_{+}$ ; the same with $H_{-} : = c - δ / 2$ and the reversed inequalities applies to $S_{-}$ . These three-segment paths are polygonal paths (Polygonal paths and polygonally connected subsets of $R^{n}$ , A finite concatenation of straight segments in $R^{n}$ is a continuous path), so a polygonally connected set is path-connected and hence connected (Every path-connected space is connected, and every path component lies inside a component).

[F7]

Rigid motions and reflections: $R = Q \cdot + b$ and $ρ (y, s) = (y, - s)$ are continuous bijections with continuous inverses and preserve distances; images of connected sets under them are connected (Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces, A continuous image of a connected space is connected, and connectedness is a topological property).

[F8]

$C^{k}$ maps and their algebra: $h \in C^{k}$ remains $C^{k}$ under negation; the coordinate functions $(y, s) \mapsto y$ and $(y, s) \mapsto s$ and affine maps are $C^{\infty}$ ; the set ${s > h (y)}$ is open as the preimage of the open set $(0, \infty)$ under the continuous map $s - h (y)$ ( $C^{k}$ Euclidean maps and diffeomorphisms, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Metric continuity characterisations, with countable choice for the sequential converse, Total differentiability gives a local $O (∥ h ∥_{2})$ increment bound and therefore continuity).

[F9]

Every connected component of an open subset of $R^{n}$ is open in $R^{n}$ and polygonally connected; components are nonempty, pairwise disjoint, cover the space, are the maximal connected subsets, and are closed; a union of connected sets with a common point is connected (Every connected component of an open subset of $R^{n}$ is open and polygonally connected, Connected components, quasicomponents, and totally disconnected spaces, The components of a space are its maximal connected subsets, they partition it, and each of them is closed, A union of connected subspaces with a point in common is connected, and so is a union of a family in which every member meets a fixed connected member).

[F10]

A nonempty space is connected exactly when its only clopen subsets are the empty set and the whole space; a connected space is not the disjoint union of two nonempty open subsets; $R^{3}$ is connected (For a topological space the following agree: no separation exists, the only clopen subsets are $\emptyset$ and $X$ , and every continuous map to the two-point discrete space is constant, Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets, $R^{n}$ is polygonally connected, connected, locally path-connected and locally connected).

Proof

technique · direct

1.1F1F2F3

Put $K : = \overline{D_{1}} \cup \dots \cup \overline{D_{m}}$ (so $K = \emptyset$ when $m = 0$ ). Each $\overline{D_{j}}$ is closed and bounded, hence compact [F1, F2]; $K$ is a finite union of compact sets, hence closed and bounded and therefore compact [F1]. By hypothesis $K \subseteq Ω$ , so $\partial Ω \cap K = \emptyset$ , and the sets $\overline{D_{1}}, \dots, \overline{D_{m}}$ are pairwise disjoint. Put $Ω_{v} : = Ω ∖ K$ ; it is open, since $Ω$ is open and $K$ is closed, and bounded, since $Ω$ is; and, when $m > 0$ , for every $x \in \partial Ω$ we have $d (x, K) > 0$ by [F3]. For $m = 0$ set $ρ_{x} : = 1$ ; for $m > 0$ set $ρ_{x} : = d (x, K) > 0$ . In either case $B (x, ρ_{x} / 2)$ misses $K$ .

2.1F2F3step 1.1

$Ω_{v} \neq \emptyset$ . Since $Ω$ is a bounded $C^{k}$ domain, $\partial Ω \neq \emptyset$ [F5]; fix $x \in \partial Ω$ and put $ρ : = ρ_{x} > 0$ as in step 1.1. Every ball about $x$ meets $Ω$ while $x \notin Ω$ [F2], so there is $y \in B (x, ρ / 2) \cap Ω$ . Then $d (y, q) \geq d (x, q) - d (x, y) > ρ - ρ / 2 > 0$ for every $q \in K$ , so $y \notin K$ ; hence $y \in Ω_{v}$ .

2.2F2F3F4F5step 1.1

$\partial Ω_{v} = \partial Ω \cup \partial D_{1} \cup \dots \cup \partial D_{m}$ as a disjoint union. For $x \in \partial D_{j}$ : $x \in \overline{D_{j}} \subseteq Ω$ , so $x \notin Ω_{v}$ ; every ball about $x$ meets $D_{j} \subseteq K$ [F2], and for a sufficiently small ball it lies in $Ω$ and misses $⋃_{i \neq j} \overline{D_{i}}$ (positive distances between the pairwise disjoint compact sets [F3]) and meets $R^{3} ∖ \overline{D_{j}}$ [F4, F5], hence meets $Ω_{v}$ ; so $x \in \partial Ω_{v}$ . For $x \in \partial Ω$ : every ball about $x$ meets $Ω$ while $x \notin K$ with $B (x, ρ_{x} / 2) \cap K = \emptyset$ [step 1.1], so small balls meet $Ω_{v}$ ; and $x \notin Ω_{v}$ ; so $x \in \partial Ω_{v}$ . Conversely let $x \in \partial Ω_{v}$ ; then $x \in \overline{Ω_{v}} \subseteq \overline{Ω}$ and $x \notin Ω_{v}$ [F2]. If $x \notin Ω$ then $x \in \partial Ω$ . If $x \in Ω$ , then $x \in K$ , say $x \in \overline{D_{j}}$ ; if $x \in D_{j}$ then $D_{j}$ is open and a small ball about $x$ lies in $D_{j} \subseteq K$ , so it misses $Ω_{v}$ , contradicting $x \in \overline{Ω_{v}}$ ; hence $x \in \overline{D_{j}} ∖ D_{j} = \partial D_{j}$ . The displayed pieces are pairwise disjoint because $K \subseteq Ω$ and the $\overline{D_{j}}$ are pairwise disjoint, and each is nonempty and compact [F5].

3.1F4F5F7F8step 1.1step 2.2

At every point of $\partial Ω_{v}$ the vacuum region $Ω_{v}$ is locally exactly the one-sided subgraph of a $C^{k}$ graph. Let $x \in \partial Ω$ . Use the chart of [F4] for $Ω$ at $x$ and let $ρ : = ρ_{x} > 0$ [step 1.1]; replacing $W$ by the open set $W \cap B (x, ρ / 2)$ preserves the chart identity [F4] and makes $W \cap K = \emptyset$ , so $Ω_{v} \cap W = Ω \cap W$ and $R (Ω_{v} \cap W) = R (W) \cap {s < h (y)}$ . Let $x \in \partial D_{j}$ . Use the chart of [F4] for $D_{j}$ at $x$ , and replace $W$ by an open ball about $x$ contained in $Ω$ and avoiding $⋃_{i \neq j} \overline{D_{i}}$ ; this keeps the chart identity [F4] and gives $Ω_{v} \cap W = W ∖ \overline{D_{j}}$ , whose image is $R (W) \cap {s > h (y)}$ : points with $s < h (y)$ lie in $D_{j}$ , points with $s = h (y)$ lie on $\partial D_{j}$ [F4, F5]. Put $ρ_{0} (y, s) : = (y, - s)$ and $R_{0} : = ρ_{0} \circ R$ , $h_{0} : = - h$ ; then $R_{0}$ is a rigid motion, $h_{0} \in C^{k}$ , and $R_{0} (Ω_{v} \cap W) = R_{0} (W) \cap {s < h_{0} (y)}$ [F7, F8].

3.2F5step 2.2

The boundary pieces of [step 2.2] are the compact embedded $C^{k}$ hypersurfaces of the boundary lemma, and they are pairwise disjoint with nonempty compact pieces; this is (b).

4.1F4F8step 1.1step 2.1step 2.2step 3.1

$Ω_{v}$ is a bounded $C^{k}$ domain: it is nonempty [step 2.1], bounded and open [step 1.1], and by [step 2.2] every boundary point is treated by one of the two local charts of [step 3.1], each exhibiting $Ω_{v}$ locally as the one-sided subgraph of a $C^{k}$ function over an open ball after a rigid motion [F4, F8]. This is (a).

4.2F4F7F8step 3.1

The normals. In the $\partial Ω$ case of [step 3.1] the chart data for $Ω_{v}$ and for $Ω$ are identical, so the prescribed outward normal of $Ω_{v}$ , computed by the formula of [F4], equals that of $Ω$ at every point of $\partial Ω$ . In the $\partial D_{j}$ case, the chart of $Ω_{v}$ is $R_{0} = ρ_{0} \circ R$ with graph $h_{0} = - h$ ; the orthogonal part of $R_{0}$ is $ρ_{0} Q$ , whose inverse is $Q^{T} ρ_{0}$ , so [F4] gives the vacuum normal $(Q^{T} ρ_{0}) (- D h_{0} (y), 1) / \sqrt{1 + ∣ D h_{0} (y) ∣^{2}} = (Q^{T} ρ_{0}) (D h (y), 1) / \sqrt{1 + ∣ D h (y) ∣^{2}}$ , and since $ρ_{0} (D h, 1) = (D h, - 1) = - (- D h, 1)$ this equals $- Q^{T} (- D h (y), 1) / \sqrt{1 + ∣ D h (y) ∣^{2}}$ , the negative of the prescribed outward normal of $D_{j}$ [F4, F8]. This is (c).

4.3F3F6F7F8step 3.1

Connected collars and patches. Fix one of the charts of [step 3.1], shrinking its neighborhood to be bounded first. Write $R$ for its rigid motion, $W$ for its domain, $h$ for its graph function and $B^{''}$ for the set of parameters $y$ with $(y, h (y)) \in R (W)$ , an open subset of $B$ containing the parameter of the distinguished boundary point. Choose a ball $B_{1}$ with $\overline{B_{1}} \subseteq B^{''}$ ; the compact graph set $G : = {(y, h (y)) : y \in \overline{B_{1}}} \subseteq R (W)$ (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset, A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value) is disjoint from the closed set $R^{3} ∖ R (W)$ , the function $z \mapsto d (z, R^{3} ∖ R (W))$ is positive and continuous on $G$ (its Lipschitz constant is at most one by the triangle inequality), hence has a positive minimum $δ$ by [F3]. Its complement is nonempty because $W$ is bounded. By continuity of $h$ at the parameter $y_{x}$ of the distinguished point choose a ball $B_{1}^{'} \subseteq B_{1}$ around $y_{x}$ with $∣ h (y) - h (y_{x}) ∣ \leq δ / 4$ for $y \in B_{1}^{'}$ [F8]. In the inner case the region $T : = {(y, s) : y \in B_{1}^{'}, h (y) < s < h (y) + δ}$ is contained in $R (W)$ (every point of $T$ is within $δ$ of its graph point, which lies in $G$ ) and in ${s > h (y)}$ , so $R^{- 1} (T) \subseteq Ω_{v} \cap W$ ; in the outer case the same holds for $T : = {(y, s) : y \in B_{1}^{'}, h (y) - δ < s < h (y)}$ and ${s < h (y)}$ , by the descriptions of [step 3.1]. The set $T$ is polygonally connected by [F6], hence connected, so $N : = R^{- 1} (T)$ is connected as its continuous image [F7]; moreover $N$ accumulates at every point $X (y)$ of the patch $P : = X (B_{1}^{'}) \subseteq \partial Ω_{v}$ , since the points $(y, h (y) \pm ε)$ with $0 < ε < δ$ lie in $T$ and converge to $(y, h (y))$ as $ε ↓ 0$ .

5.1F9step 4.1

Each connected component of the open set $Ω_{v}$ is open in $R^{3}$ and polygonally connected [F9]; in particular every component is open.

5.2F2F9step 3.1step 4.3

Each patch is incident to exactly one component. The connected nonempty collar $N$ of step 4.3 lies in one component $U$ of $Ω_{v}$ [F9]. For $z = X (y_{0}) \in P$ , continuity of $h$ and openness of $B_{1}^{'}$ give a ball $B (z, r)$ so small that its chart coordinates satisfy $y \in B_{1}^{'}$ and $∣ s - h (y) ∣ < δ$ , and that $B (z, r) \subset W$ . By step 3.1 the vacuum portion of this ball lies in the one-sided collar, hence $B (z, r) \cap Ω_{v} \subset N \subset U$ . If $z \in \partial U_{1}$ for another component, the ball meets $U_{1}$ by [F2], so $U_{1}$ meets $N$ and equals $U$ by [F9]. Conversely, every $z \in P$ is in $\partial U$ because the collar accumulates there and $z \notin Ω_{v}$ . Thus every point of $P$ is incident to exactly $U$ .

6.1F2F9F10step 2.2step 3.2step 5.2

Every component is incident to a patch. Let $U$ be a component of $Ω_{v}$ . It is nonempty, open and bounded, so $\partial U \neq \emptyset$ : otherwise $\overline{U} = U$ , making $U$ a nonempty clopen proper subset of the connected space $R^{3}$ (proper since $U$ is bounded), contradicting [F10]. Also $\partial U \subseteq \partial Ω_{v}$ : if $x \in \partial U$ then $x \in \overline{U} \subseteq \overline{Ω_{v}}$ and $x \notin Ω_{v}$ , because $Ω_{v}$ is open and $x$ lies neither in the open set $U$ nor in its open complement in $Ω_{v}$ [F2, F9]. By [step 2.2] and [step 3.2] every point of $\partial Ω_{v}$ lies on one of the finitely many boundary pieces, and the patches of the kind constructed in [step 4.3] cover each piece; so some such patch $P$ meets $\partial U$ , and [step 5.2] identifies $U$ as the unique component incident to $P$ .

7.1step 3.2step 5.2step 6.1

Finitely many components. The patches of the kind constructed in [step 4.3], for all boundary points, cover $\partial Ω_{v}$ [step 6.1], and $\partial Ω_{v}$ is compact [step 3.2]; hence finitely many of them already cover $\partial Ω_{v}$ . Each determines exactly one component [step 5.2], and every component is determined by one of them [step 6.1]; therefore the components of $Ω_{v}$ are among these finitely many, so $Ω_{v}$ has finitely many connected components.

8.1F4F8F9F10step 1.1step 3.1step 4.2step 5.1step 3.2step 4.3step 5.2step 6.1step 7.1∎

Each component is a bounded $C^{k}$ domain with the stated boundary. Let $U$ be a component of $Ω_{v}$ and let $P$ be a patch incident to $U$ [step 6.1]. The local description of [step 3.1] shows that, in the chart of $P$ , $U$ coincides with the one-sided region described there, hence is locally exactly the one-sided subgraph of the $C^{k}$ function $h$ (outer case) or of the reflected $C^{k}$ function $h_{0} = - h$ (inner case) over a ball [F4, F8, step 3.1, step 5.2]. So $U$ is a bounded $C^{k}$ domain: it is open [step 5.1], bounded (a subset of bounded $Ω_{v}$ ) and nonempty [F9]. Every point of a patch incident to $U$ is a boundary point of $U$ (it lies in $\partial Ω_{v}$ and in the closure of the collar $N \subseteq U$ [step 4.3, step 5.2]), and conversely every boundary point of $U$ lies in a patch incident to $U$ [step 6.1]; hence $\partial U$ equals the union of those patches in a finite covering that are incident to $U$ . It is compact because it is closed in $R^{3}$ [F2] and contained in the compact $\partial Ω_{v}$ [step 3.2], by [F1]. Finally, let $C$ be a connected component of $\partial Ω_{v}$ ; the assignment to $x \in C$ of the component of $Ω_{v}$ incident to a patch containing $x$ takes values in a finite set and is locally constant on $C$ (near a point of a patch it is given by the single component of that patch [step 5.2]), so it is constant on the connected set $C$ [F10]; hence each boundary component of $\partial Ω_{v}$ is incident to exactly one component of $Ω_{v}$ , and $\partial U$ is a union of such boundary components. The normal formulas of [step 4.2] apply verbatim to each piece of $\partial U$ , since they are the same local charts. This is (d); connectedness of $Ω_{v}$ itself is neither assumed nor asserted.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

Uniqueness for harmonic functions on a punctured half-space with zero trace, decaying at the pole and at infinity

Statement

Let $H : = {x \in R^{3} : x_{3} > 0}$ be the open upper half-space, let $a > 0$ and $p : = a e_{3} \in H$ , where $e_{3} : = (0, 0, 1)$ . Let $w : \overline{H} ∖ {p} \to R$ be a function (closure as in Interior, closure, boundary, limit point, isolated point and dense subset of a metric space) with:

(i) $w$ is continuous (Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form) on $\overline{H} ∖ {p}$ and of class $C^{2}$ with $Δ w = 0$ on $H ∖ {p}$ (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, $C^{k}$ Euclidean maps and diffeomorphisms);

(ii) $w = 0$ on $\partial H = {x_{3} = 0}$ ;

(iii) $∣ x - p ∣ w (x) \to 0$ as $x \to p$ , $x \neq p$ ;

(iv) $w (x) \to 0$ as $∣ x ∣ \to \infty$ within $H$ : for every $ε > 0$ there is $R > 0$ with $∣ w (x) ∣ \leq ε$ for all $x \in H ∖ {\overline{B}}_{R} (0)$ .

Then $w = 0$ on $H ∖ {p}$ .

The conclusion is a uniqueness statement for the stated class only: no removable-singularity theorem, no pole condition beyond the sub-Coulomb growth (iii), no boundary regularity of $\partial H$ beyond the flat graph, and no claim about $w$ outside $H$ are asserted.

Facts & Assumptions

Given: The open upper half-space $H = {x_{3} > 0}$ , the point $p = a e_{3}$ with $a > 0$ , and a function $w : \overline{H} ∖ {p} \to R$ satisfying (i)-(iv) of the statement. Write $B_{r} (q)$ for the open ball and use the Euclidean norm and metric on $R^{3}$ .

[F1]

The weak maximum principle: if $Ω \subseteq R^{n}$ , $n \geq 2$ , is bounded, nonempty and open and $u \in C^{2} (Ω) \cap C (\overline{Ω})$ satisfies $Δ u \geq 0$ on $Ω$ , then $\max_{\overline{Ω}} u = \max_{\partial Ω} u$ (Weak maximum principle for the laplacian).

[F2]

A $C^{2}$ function is harmonic where $Δ u = 0$ ; $Δ$ is defined as the sum of the pure second partial derivatives, is linear on $C^{2}$ functions and annihilates constants (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, $C^{k}$ Euclidean maps and diffeomorphisms).

[F3]

For $n = 3$ the fundamental solution is $Φ (x) = ∣ x ∣^{- 1} / ω_{2}$ with $ω_{2} > 0$ , and for every pole $q$ the function $x \mapsto Φ (x - q)$ is smooth and harmonic on $R^{3} ∖ {q}$ ; the function $b_{q} (x) : = ∣ x - q ∣^{- 1} = ω_{2} Φ (x - q)$ is therefore a positive constant multiple of a harmonic function, hence $C^{\infty}$ and harmonic off $q$ , and it is continuous wherever it is defined (Fundamental solution for the positive operator minus Laplacian, The Laplace fundamental solution is harmonic off its pole, The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field).

[F4]

An open ball is open, its closure is the corresponding closed ball, and $x \mapsto ∣ x - q ∣$ is continuous for the Euclidean metric; the reciprocal $t \mapsto 1 / t$ is continuous on $t > 0$ ; a function with $∣ x - q ∣ \geq δ > 0$ on its domain has $∣ b_{q} (x) - b_{q} (y) ∣ \leq ∣ x - y ∣ / δ^{2}$ there, by the reverse triangle inequality (Open ball, closed ball and sphere in a metric space, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric, The finite and reverse triangle inequalities for a norm; and for $n \geq 1$ every norm $N$ on $R^{n}$ satisfies $N (x) \leq C ∥ x ∥_{1}$ and is Lipschitz, hence continuous, for $d_{2}$ , Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form, A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions).

[F5]

Closure is the smallest closed superset: if $A \subseteq F$ and $F$ is closed then $\overline{A} \subseteq F$ ; closed balls and spheres in $R^{3}$ are compact, hence closed (The closure of a nonempty $A$ is ${x : d (x, A) = 0}$ , equals $A$ together with its limit points, and is the smallest closed superset, For $n \geq 1$ , every Euclidean closed ball and every Euclidean sphere of positive radius is compact).

[F6]

The boundary of the open set $U$ is $\partial U = \overline{U} ∖ U$ ; for an open set the interior is the set itself (Interior, closure, boundary, limit point, isolated point and dense subset of a metric space).

[F7]

Unwinding hypothesis (iii): for every $ε > 0$ there is $δ > 0$ with $∣ w (x) ∣ \leq ε / ∣ x - p ∣$ for all $x$ in $\overline{H} ∖ {p}$ with $0 < ∣ x - p ∣ < δ$ (Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form).

[F8]

Hypothesis (iv) says exactly that for every $ε > 0$ there is $ρ > 0$ with $∣ w (x) ∣ \leq ε$ for all $x \in H$ with $∣ x ∣ > ρ$ (Open ball, closed ball and sphere in a metric space).

[F9]

Countable Choice holds in the ambient measure conventions inherited from the kernel supplier; the argument below is pointwise and uses no choice principle (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

1.1givenF3F7F8

Fix $y \in H ∖ {p}$ and $η > 0$ . By [F8] there is $ρ > 0$ with $∣ w ∣ \leq η$ on $H \cap {∣ x ∣ > ρ}$ ; put $R : = \max {ρ, 2 a, ∣ y ∣} + 1$ , so that $R > ∣ p ∣ + a$ , $R > ∣ y ∣$ and $∣ w ∣ \leq η$ on $H \cap {∣ x ∣ \geq R}$ . By [F7] there is $δ_{0} > 0$ with $∣ w (x) ∣ \leq η / ∣ x - p ∣$ whenever $0 < ∣ x - p ∣ < δ_{0}$ ; put $δ : = \frac{1}{2} \min {δ_{0}, a / 2, ∣ y - p ∣, R - ∣ p ∣}$ , so $0 < δ < a / 2$ , $δ < ∣ y - p ∣$ , $δ < R - ∣ p ∣$ and $δ < δ_{0}$ . Define the open set $U : = H \cap B_{R} (0) \cap {x : ∣ x - p ∣ > δ}$ , the barrier $b (x) : = ∣ x - p ∣^{- 1}$ , and $v_{\pm} : = \pm w - η (1 + b)$ on $U$ .

2.1F4F5F8step 1.1

$U$ is bounded, nonempty and open, and $y \in U$ ; explicitly, $R > ∣ y ∣$ and $δ < ∣ y - p ∣$ put $y$ in $U$ . The closure satisfies $\overline{U} \subseteq \overline{H} \cap \overline{B_{R} (0)} \cap {x : ∣ x - p ∣ \geq δ}$ : each of $\overline{H}$ , $\overline{B_{R} (0)}$ and ${x : ∣ x - p ∣ \geq δ}$ is closed (the last is the complement of the open ball $B_{δ} (p)$ ) and contains $U$ , so [F5] applies factor by factor. In particular $∣ x - p ∣ \geq δ > 0$ on $\overline{U}$ , so $p \notin \overline{U}$ and $\overline{U} \subseteq \overline{H} ∖ {p}$ .

3.1F2F3F4step 1.1step 2.1

Each $v_{\pm}$ lies in $C^{2} (U) \cap C (\overline{U})$ and satisfies $Δ v_{\pm} = 0$ on $U$ . Indeed $U \subseteq H ∖ {p}$ by construction, where $w$ is $C^{2}$ and $Δ w = 0$ , while $b = ω_{2} Φ (\cdot - p)$ is $C^{\infty}$ and harmonic off $p$ [F3]; constants are $C^{\infty}$ ; $Δ$ is linear and annihilates constants [F2], so $Δ v_{\pm} = \pm Δ w - η (0 + Δ b) = 0$ . Continuity on $\overline{U}$ holds because $w$ is continuous there by (i) and step 2.1, and $b$ is continuous on $\overline{U}$ by [F3] and [F4], the bound $∣ x - p ∣ \geq δ$ giving $∣ b (x) - b (y) ∣ \leq ∣ x - y ∣ / δ^{2}$ ; sums and scalar multiples of continuous functions are continuous [F4].

3.2F6givenstep 1.1step 2.1

Each $v_{\pm}$ is nonpositive on $\partial U$ . Let $x \in \partial U = \overline{U} ∖ U$ [F6]. Since $x \in \overline{U}$ and $x \notin U$ , at least one of $x_{3} = 0$ , $∣ x ∣ = R$ , $∣ x - p ∣ = δ$ holds, by the description of $\overline{U}$ in step 2.1 and the definition of $U$ . If $x_{3} = 0$ then $w (x) = 0$ by (ii), so $v_{\pm} (x) = - η (1 + b (x)) < 0$ . Otherwise $x_{3} > 0$ , and since at least one of the three conditions holds, $∣ x ∣ = R$ or $∣ x - p ∣ = δ$ (or both); if $∣ x ∣ = R$ then $∣ w (x) ∣ \leq η$ by the choice of $R$ in step 1.1, so $v_{\pm} (x) \leq η - η (1 + b (x)) = - η b (x) < 0$ , and if $∣ x - p ∣ = δ$ then $∣ w (x) ∣ \leq η / δ = η / ∣ x - p ∣$ by the choice of $δ$ in step 1.1 (and $x \in \overline{H} ∖ {p}$ by step 2.1), so $v_{\pm} (x) \leq η b (x) - η (1 + b (x)) = - η < 0$ .

4.1F1step 3.1step 3.2

The weak maximum principle applies to each $v_{\pm}$ on the bounded nonempty open set $U$ [step 2.1, step 3.1] and gives $\max_{\overline{U}} v_{\pm} = \max_{\partial U} v_{\pm} \leq 0$ by step 3.2, so $v_{\pm} \leq 0$ on $\overline{U}$ . In particular at $y \in U \subseteq \overline{U}$ , $\pm w (y) \leq η (1 + ∣ y - p ∣^{- 1})$ .

5.1givenF9step 1.1step 4.1∎

Since $η > 0$ was arbitrary in step 1.1 and the bound of step 4.1 holds for every $η > 0$ , the real number $t : = ∣ w (y) ∣$ satisfies $t \leq η (1 + ∣ y - p ∣^{- 1})$ for every $η > 0$ ; if $t > 0$ then $η : = t / (2 (1 + ∣ y - p ∣^{- 1})) > 0$ gives $t \leq t / 2 < t$ , a contradiction, so $t = 0$ , that is $w (y) = 0$ . As $y \in H ∖ {p}$ was arbitrary, $w = 0$ on $H ∖ {p}$ . This proves uniqueness in the stated sub-Coulomb class; no removable-singularity theorem, no boundedness or growth assumption beyond (iii)-(iv), and no boundary regularity beyond the flat trace (ii) was used, and the only ambient choice convention is the one inherited from the kernel interfaces [F9].

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The ideal electrostatic conductor model

Physical scope

The ideal electrostatic conductor idealization of the classical framework in one named inertial frame: finitely many compact smooth field-free conductor bodies inside a general open laboratory region, with a separate bounded smooth ambient branch, all conductor charge carried by continuous stationary surface densities, a connected vacuum region in which the pure electrostatic regime and the adopted distributional Maxwell equations hold, and a continuous equilibrium potential that is constant on each conductor body and continuous across its surface. The item is a model choice: it asserts no existence, uniqueness, dynamics, conductivity, forces, energy, radiation, measurement or apparatus claim, and it excludes cavity configurations with disconnected vacuum regions.

Postulate

Under the Countable Choice convention of the stated boundary and sheet-distribution interfaces (The Axiom of Countable Choice ( ${A C}_{ω}$ )), adopt as a physical model of ideal electrostatic conductors, for the classical laboratory framework of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields with the units and constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values: a conductor configuration consists of the following data, and every model of this postulate is required to satisfy the clauses below. No experiment is reported and no empirical value is inferred; this is an explicitly adopted equilibrium idealization together with its geometric and source conventions.

(a) [Geometry] a laboratory region $Ω \times I$ with $Ω \subseteq R^{3}$ an arbitrary open ambient region, a finite family $D_{1}, \dots, D_{m}$ ( $m \geq 1$ ) of bounded $C^{\infty}$ domains with pairwise disjoint closures in the sense of Bounded C^k domains and boundary charts with closures strictly inside $Ω$ (the conductor bodies), the conductor $D : = D_{1} \cup \dots \cup D_{m}$ , its surface $\partial D$ , and the vacuum region $Ω_{v} : = Ω ∖ \overline{D}$ . By The boundary of a bounded smooth domain is a compact embedded hypersurface, $\partial D = \partial D_{1} \cup \dots \cup \partial D_{m}$ is a disjoint union of compact embedded $C^{\infty}$ hypersurfaces. The vacuum $Ω_{v}$ is open, and near each conductor surface it is locally the smooth exterior side of that conductor. No boundedness or regularity of the ambient boundary is inferred from openness. In the additional bounded smooth ambient branch, assume that $Ω$ is a bounded $C^{\infty}$ domain; only then The vacuum region around compact smooth conductors is a bounded smooth domain gives that $Ω_{v}$ is a bounded $C^{\infty}$ domain with the specified outer/conductor boundary charts; we additionally require $Ω_{v}$ to be connected (configurations whose vacuum region is separated by the conductors, for instance by an enclosing closed shell, are outside this model as formulated here).

(b) [Equilibrium and sources] the configuration is in the pure electrostatic regime of The static and pure electrostatic regimes and their fixed-time reading on the vacuum region: $B = 0$ , $J = 0$ , $E$ is the fixed-time electric field of The electric and magnetic fields as fixed-time spatial vector fields, the volume source density $ρ_{v}$ is a continuous real map on $\overline{Ω_{v}}$ (C m $^{- 3}$ , Charge parameter, frame charge and current densities, volume charge and flux, and the four-current) extended by zero to $D$ ; the conductor charge is a stationary surface charge, i.e. a continuous density $σ : \partial D \to R$ (C m $^{- 2}$ ) whose pairing with test functions is the stationary-sheet source of Distributional Maxwell equations with point and stationary-sheet sources(c) with tangential current $K = 0$ and $S = \partial D$ ; and the field pair $(E, 0)$ together with the source pair $(ρ_{v} + σ δ_{\partial D}, 0)$ satisfies the adopted distributional vacuum Maxwell equations Distributional vacuum Maxwell equations for the declared point and stationary-sheet sources. Inside the conductor the field vanishes: $E \equiv 0$ on $D$ (the equilibrium condition); equivalently the conductor interior is field-free and carries no volume charge.

(c) [Potential and equipotential convention] there is a function $φ \in C^{0} (Ω) \cap C^{2} (Ω_{v})$ with $E = - \nabla φ$ on $Ω_{v}$ and with $φ$ constant on each conductor body, $φ ∣_{D_{j}} \equiv V_{j}$ for real constants $V_{j}$ (volts, Framework-local SI units, the vacuum constants and the exact/measured status of their values); by continuity of $φ$ the trace of $φ$ on each $\partial D_{j}$ from the vacuum side equals $V_{j}$ as well. This is the single-function convention of an ideal conductor with no double layer: the model reads the same potential value on both sides of $\partial D_{j}$ , while the normal component of $E$ may jump.

(d) [Data] a model is completed either by the component potentials $V_{1}, \dots, V_{m}$ together with the volume source $ρ_{v}$ and the data on $\partial Ω$ where the laboratory region is bounded, or by the component total charges $Q_{j} : = \int_{\partial D_{j}} σ d S$ (C) together with those same ambient data. These are two declared data types; the postulate asserts neither that both data types are simultaneously specifiable nor that either is solvable for arbitrary data.

(e) [Scope] This is a static, frame-relative equilibrium idealization: it excludes transient finite-conductivity dynamics, relaxation to equilibrium, moving surface charge, electrode polarization and contact effects, and it is not derived from a microscopic matter model. It asserts no existence or uniqueness theorem (those are separate conditional results), no force, no energy, no radiation, no measurement and no apparatus realization; the value $V_{j}$ is a model parameter and the surface density $σ$ is an idealization variable determined by the equilibrium problem when it is solvable. The general open ambient branch retains compact conductor bodies in exterior configurations. Its additional bounded smooth ambient branch is distinct from the separately declared grounded-half-space and infinite parallel-plane idealizations, whose conductor bodies are themselves noncompact. Those examples retain the equilibrium/equipotential interpretation but their noncompact bodies and plane distributions are not instances of this compact-body configuration or its compact-sheet predicate. No arbitrary exterior solvability is asserted.

Interpretation and source basis

The postulate records a model choice, not a deduction, and its clauses are the exact objects the rest of this page consumes. The field-free interior with all charge on the surface and the constant-potential (equipotential) surface are the standard ideal-conductor clauses of the source treatments (Griffiths §2.5.1 printed pp. 96-98; Tong §2.4 printed pp. 30-31; Chew §32.1-32.2 printed pp. 351-354), presented there for equilibrium configurations and reproduced here as adopted assumptions rather than as derived theorems. The division into two data types in (d) is the fixed-potential versus fixed-charge distinction of Tong §2.4.2 printed pp. 33-35 and Griffiths §3.1.5-3.1.6 printed pp. 116-121; it is a definitional choice about which quantities are prescribed. Nothing in (e) turns the idealization into a universal statement about real metals: finite conductivity, transients, contact potentials, surface forces and material properties are outside the model, and no numerical or empirical value of any material quantity is asserted.

The mathematical content used to name the objects is exact and proved earlier on this page: the surface $\partial D$ of finitely many pairwise disjoint bounded $C^{\infty}$ bodies is a disjoint finite union of compact embedded smooth hypersurfaces, and in the bounded smooth ambient branch the vacuum region is again a bounded smooth domain whose boundary pieces are the outer boundary and the conductor surfaces. The stationary-sheet pairing $σ δ_{\partial D}$ is the surface-distribution object of Surface distributions and interfacial jumps as used by Distributional Maxwell equations with point and stationary-sheet sources(c), and the solution predicate for the field-source pair is the separately adopted Distributional vacuum Maxwell equations for the declared point and stationary-sheet sources; the postulate itself neither proves that predicate nor asserts that it is satisfiable for the declared data. The countable-choice convention belongs to those measure and distribution interfaces and is carried here explicitly (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

The model is deliberately silent about configurations whose conductor bodies enclose a cavity of the vacuum: the bounded smooth ambient branch requires $Ω_{v}$ connected, so configurations separated by the conductors are outside the model as formulated. The separately declared grounded-plane and ideal parallel-plate examples are noncompact idealizations and are not instances of this compact-body configuration; they retain their own explicit statements and are not theorems about this postulate.

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Surface charge and normal field on an ideal conductor

Physical scope

The ideal-conductor boundary conditions of classical electrostatics in a named inertial frame: for a stationary smooth conductor surface in the pure electrostatic regime and fields with the stated one-sided regularity, the surface charge density equals epsilon_0 times the normal vacuum-side field, the vacuum-side field is normal to the surface, the potential trace is constant on each connected piece of the conductor surface with the model's component value, and the total component charges are the outward flux integrals of the vacuum field. The statement is conditional and pointwise; it excludes moving or nonsmooth surfaces, magnetic and current sources, forces, energy, material response and any empirical claim.

Statement

Assume the Axiom of Countable Choice ( ${A C}_{ω}$ ) inherited from the surface and distribution interfaces (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $(Ω; D_{1}, \dots, D_{m}; Ω_{v}; E; ρ_{v}; σ; V_{j})$ be a model of The ideal electrostatic conductor model with surface $S : = \partial D$ and vacuum region $Ω_{v}$ , with the fields of The electric and magnetic fields as fixed-time spatial vector fields, the constants and units of Framework-local SI units, the vacuum constants and the exact/measured status of their values and the pure electrostatic regime of The static and pure electrostatic regimes and their fixed-time reading carried by the model, and suppose additionally that the field $E$ satisfies the joint one-sided $C^{1}$ regularity hypothesis of Stationary vacuum interface jumps and the absence of magnetic sheet sources across $S$ : around every $x_{0} \in S$ each side of $S$ is the restriction of a $C^{1}$ field on an open neighbourhood of the corresponding closed side of a smaller graph chart. Let $n$ denote the unit normal on $S$ pointing from the conductor side into the vacuum side (equivalently: $n$ is the inward normal of $Ω_{v}$ and the outward normal of $D$ on $S$ , by the local exterior conductor chart; in the bounded smooth ambient branch this is the sign convention of The vacuum region around compact smooth conductors is a bounded smooth domain(c)), and let $E_{+}$ denote the vacuum-side trace of $E$ (the field is zero on the conductor side, so $[E] = E_{+}$ ). Then, pointwise on $S$ :

(a) [Normal sheet charge] $n \cdot E_{+} = σ / ε_{0}$ and $n \times E_{+} = 0$ ; equivalently $σ = ε_{0} n \cdot E_{+}$ and the tangential part of the vacuum-side field vanishes: $E_{+}$ has no component tangent to $S$ . In particular a conductor surface with $σ = 0$ at a point has $E_{+} = 0$ there (both traces vanish), and the sign of $σ$ is the sign of the outward normal field component.

(b) [Normal derivative of the potential] with $φ$ the model's potential and $\partial φ / \partial n : = \nabla φ_{+} \cdot n$ the one-sided normal derivative in the direction $n$ from the conductor into the vacuum (Classical normal derivative), one has

$\frac{\partial φ}{\partial n} = - \frac{σ}{ε_{0}}, equivalently σ = - ε_{0} \frac{\partial φ}{\partial n} .$

(c) [Equipotential boundary traces] the trace $φ_{+}$ of $φ$ on $S$ is constant on each connected component of $S$ , with value $V_{j}$ on every connected component contained in $\partial D_{j}$ ; consequently the boundary value problem in the vacuum region is a Dirichlet problem whose data are the constants $V_{j}$ on the conductor surfaces.

(d) [Total charges] the total charge of conductor $j$ is

$Q_{j} = \int_{\partial D_{j}} σ d S = ε_{0} \int_{\partial D_{j}} n \cdot E_{+} d S = ε_{0} \int_{\partial D_{j}} \frac{\partial φ}{\partial ν} d S,$

where $ν$ is the local outward normal of the vacuum along the smooth conductor interface $S$ , without asserting regularity of a general ambient outer boundary (so $ν = - n$ on $\partial D$ and the last equality uses $\partial φ / \partial ν = - \partial φ / \partial n$ ); the surface integrals exist as integrals of continuous functions over the compact embedded hypersurfaces $\partial D_{j}$ (Surface integration on compact C1 hypersurfaces, The boundary of a bounded smooth domain is a compact embedded hypersurface).

(e) [Data reading] the two data types of the model are related through (d): if the component potentials and ambient data are fixed and the equilibrium solution is given, the total charges are determined by (d); prescribing the total charges instead determines the potentials only through the (separate) equilibrium problem, and this theorem asserts no existence or uniqueness for charge data.

(f) [Scope] The statement is pointwise and stationary: it assumes the one-sided joint regularity above, a $C^{\infty}$ conductor surface, and the pure electrostatic regime; it excludes moving surface charge, normal sheet currents, magnetic sheet sources, nonsmooth or cornered conductors, fields without one-sided traces, forces, energy and any measurement; no self-force or material response is asserted.

Facts & Assumptions

Given: A model of The ideal electrostatic conductor model with conductor surface $S = \partial D = \partial D_{1} \cup \dots \cup \partial D_{m}$ , vacuum region $Ω_{v}$ , field $E$ with $E \equiv 0$ on $D$ , continuous stationary surface density $σ$ on $S$ , potential $φ \in C^{0} (Ω) \cap C^{2} (Ω_{v})$ with $E = - \nabla φ$ on $Ω_{v}$ and $φ \equiv V_{j}$ on $D_{j}$ , and the joint one-sided $C^{1}$ regularity of the statement across $S$ .

[F1]

The conductor model: $E \equiv 0$ on $D$ ; the pair $(E, 0)$ with the source pair $(ρ_{v} + σ δ_{S}, 0)$ , $K = 0$ , satisfies the adopted distributional vacuum Maxwell equations Distributional vacuum Maxwell equations for the declared point and stationary-sheet sources; $σ$ is the stationary-sheet source of Distributional Maxwell equations with point and stationary-sheet sources(c) under Surface distributions and interfacial jumps; $φ$ is continuous on $Ω$ and $C^{2}$ on $Ω_{v}$ with $E = - \nabla φ$ there, constant on each $D_{j}$ with value $V_{j}$ (The ideal electrostatic conductor model, The static and pure electrostatic regimes and their fixed-time reading, The electric and magnetic fields as fixed-time spatial vector fields, Framework-local SI units, the vacuum constants and the exact/measured status of their values).

[F2]

The fixed-vacuum-jump theorem: under ${A C}_{ω}$ , the stationary compact embedded $C^{2}$ interface hypotheses, the joint one-sided $C^{1}$ field regularity and the weak Maxwell equations with the stationary-sheet source of [F1], one has pointwise on $S$ $n \cdot [E] = σ / ε_{0}$ , $n \times [E] = 0$ , $n \cdot [B] = 0$ , $n \times [B] = μ_{0} K$ , with $[F] = F_{+} - F_{-}$ and $n$ pointing from the $Ω_{-}$ side to the $Ω_{+}$ side (Stationary vacuum interface jumps and the absence of magnetic sheet sources, Surface distributions and interfacial jumps).

[F3]

For a bounded $C^{\infty}$ conductor surface the outward normal of the vacuum region on $\partial D_{j}$ is $n_{j} = - n$ (the negative of the conductor's outward normal) whenever the bounded smooth ambient hypothesis of The vacuum region around compact smooth conductors is a bounded smooth domain(c) is in force; the local exterior conductor chart of the model identifies the vacuum-side normal with the conductor's outward normal in general (The ideal electrostatic conductor model(a), The vacuum region around compact smooth conductors is a bounded smooth domain(c)).

[F4]

Classical one-sided normal derivative: for the vacuum side, $\partial φ / \partial n : = \nabla φ_{+} \cdot n$ is the normal derivative of the $C^{1}$ vacuum-side restriction $φ$ in the direction $n$ , where $φ$ is $C^{2}$ on the vacuum side and its gradient has the continuous trace $- E_{+}$ (Classical normal derivative, $C^{k}$ Euclidean maps and diffeomorphisms).

[F5]

$\partial D$ is a finite disjoint union of compact embedded smooth hypersurfaces carrying a finite surface measure on which continuous functions are integrable (The boundary of a bounded smooth domain is a compact embedded hypersurface, Surface integration on compact C1 hypersurfaces).

[F6]

Euclidean vector decomposition: a vector $v$ with $n \cdot v = 0$ and $n \times v = 0$ for the unit vector $n$ is $v = 0$ , since $v = (n \cdot v) n + n \times (v \times n)$ for orthonormal triples (The electric and magnetic fields as fixed-time spatial vector fields, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The cross product in $R^{3}$ ).

Proof technique: direct.

Proof

1.1F1F2F6given

By [F2] with the conductor side as $Ω_{-}$ and the vacuum side as $Ω_{+}$ (the normal $n$ of the statement points from the conductor into the vacuum, matching [F2]), the jump identities $n \cdot [E] = σ / ε_{0}$ and $n \times [E] = 0$ hold pointwise on $S$ ; the conductor-side trace vanishes because $E \equiv 0$ on $D$ [F1], so $[E] = E_{+} - 0 = E_{+}$ . This gives $n \cdot E_{+} = σ / ε_{0}$ and $σ = ε_{0} n \cdot E_{+}$ , and $n \times E_{+} = 0$ , which is (a); if $σ = 0$ at a point then $n \cdot E_{+} = 0$ and $n \times E_{+} = 0$ there, so $E_{+} = 0$ by [F6], and the sign of $σ$ is the sign of $n \cdot E_{+}$ since $ε_{0} > 0$ .

1.2F1F5given

Equipotential boundary traces. For $x \in \partial D_{j}$ , the boundary definition supplies points of $D_{j}$ arbitrarily close to $x$ . Since $φ$ is continuous on $Ω$ and $φ ∣_{D_{j}} = V_{j}$ [F1], its value and vacuum trace at $x$ equal $V_{j}$ . Thus the trace is constant on all of $\partial D_{j}$ , and in particular on every connected component contained there. The separated boundary pieces partition $S$ [F5], so this proves (c) directly from the model's continuous equipotential convention.

2.1F1F4step 1.1

On the vacuum side $E = - \nabla φ$ [F1], so $\nabla φ_{+} = - E_{+}$ [F4] and $\partial φ / \partial n = \nabla φ_{+} \cdot n = - E_{+} \cdot n = - σ / ε_{0}$ by [step 1.1]; this is (b).

3.1F3F4F5step 1.1step 2.1

Total charges. The surface integrals in $Q_{j} = \int_{\partial D_{j}} σ d S$ exist as integrals of the continuous density $σ$ over the compact embedded hypersurface $\partial D_{j}$ with finite surface measure [F5]. Substituting $σ = ε_{0} n \cdot E_{+}$ from [step 1.1] gives the middle expression. For the last expression, on $\partial D_{j}$ the local outward normal of the vacuum is $ν = - n$ [F3], so $\partial φ / \partial ν = \nabla φ_{+} \cdot ν = - \nabla φ_{+} \cdot n = - \partial φ / \partial n = σ / ε_{0}$ by [step 2.1]; integrating over $\partial D_{j}$ gives $Q_{j} = ε_{0} \int_{\partial D_{j}} \partial φ / \partial ν d S$ . This is (d).

4.1F1F2F5givenstep 3.1∎

Data reading and scope. Clause (e) follows from the definitions in (d): the total charges are determined by the equilibrium solution through the displayed identities, while charge data enter only through a separate equilibrium problem, for which this theorem proves neither existence nor uniqueness [given]. The statement is pointwise on $S$ and stationary: no moving charge, normal sheet current, magnetic sheet source, corner or field without one-sided traces is considered, no product of distributions and no corner equation is formed, and no force, energy, radiation, self-force, material response or measurement is asserted [F1, F2]; the surface integrals exist but no regularity of a general ambient outer boundary is claimed [F5].

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Uniqueness and flux compatibility for the electrostatic boundary value problem

Physical scope

Uniqueness for the classical and weak Dirichlet/Neumann electrostatic boundary value problems in a bounded vacuum region of a named inertial frame, with the componentwise Neumann freedom, the necessary flux compatibility, and the conductor-potential reading: identical complete Dirichlet data determine the potential and field exactly; compatible Neumann data determine the field and leave one potential constant per connected component until a normalization is fixed. The statement is conditional on the stated boundary, decay and regularity data and on the cited mathematical suppliers; it asserts no existence, no arbitrary-boundary solvability, no material interfaces beyond the declared sheet branch, and no empirical claim.

Statement

Assume the Axiom of Choice and Countable Choice as declared by the cited weak-solvability and trace suppliers (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $Ω_{v} \subset R^{3}$ be a bounded $C^{\infty}$ domain with finitely many connected components (the vacuum region of the additional bounded $C^{\infty}$ ambient branch of The ideal electrostatic conductor model by The vacuum region around compact smooth conductors is a bounded smooth domain, or any bounded smooth domain), let $ρ$ be continuous on $\overline{Ω_{v}}$ (the fixed-time volume source density, C m $^{- 3}$ of Framework-local SI units, the vacuum constants and the exact/measured status of their values), and let $φ_{1}, φ_{2} \in C^{2} (\overline{Ω_{v}})$ be real potentials of the pure electrostatic regime The static and pure electrostatic regimes and their fixed-time reading with fields $E_{i} : = - \nabla φ_{i}$ of The electric and magnetic fields as fixed-time spatial vector fields and

$- Δ φ_{i} = ρ / ε_{0} pointwise on Ω_{v}, Δ = div \nabla .$

The Laplacian convention is The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field. Let $ν$ denote the outward unit normal of $Ω_{v}$ (Classical normal derivative). Then:

(a) [Classical Dirichlet uniqueness] if $φ_{1} = φ_{2}$ on $\partial Ω_{v}$ , then $φ_{1} = φ_{2}$ on $\overline{Ω_{v}}$ , and hence $E_{1} = E_{2}$ . The conclusion holds component by component; each connected component of a bounded $C^{\infty}$ domain is again a bounded $C^{\infty}$ domain, to which the connected-domain uniqueness corollary applies.

(b) [Classical Neumann uniqueness and compatibility] if $\partial_{ν} φ_{1} = \partial_{ν} φ_{2}$ on $\partial Ω_{v}$ , then $φ_{1} - φ_{2}$ is constant on each connected component of $Ω_{v}$ , and conversely adding one real constant per component to a solution preserves both the equation and the Neumann data; hence the field $E$ is unique while the potential is determined only up to componentwise constants. Moreover the Neumann data are necessarily compatible:

$\int_{Ω_{v}^{(j)}} ρ d x + ε_{0} \int_{\partial Ω_{v}^{(j)}} \partial_{ν} φ d S = 0 for every connected component Ω_{v}^{(j)} .$

This compatibility is necessary, not sufficient, and is exactly the flux identity of the divergence theorem for the Poisson equation.

(c) [Weak variants] (i) if $g \in H^{1 / 2} (\partial Ω_{v})$ (The fractional Sobolev space on a compact $C^{1}$ boundary) is prescribed as a trace and $φ_{1}, φ_{2} \in H^{1} (Ω_{v})$ are weak solutions of the Dirichlet problem with the same trace $g$ , then $φ_{1} = φ_{2}$ ; (ii) if the conormal data are prescribed compatibly, weak $H^{1}$ solutions are unique modulo constants on each connected component, and for connected $Ω_{v}$ this is the normalised solution of Weak Neumann and transmission existence with interface charges; (iii) [sheet/transmission variant] for the finite disjoint smooth interfaces and outer Dirichlet data of Weak and classical solvability of the elliptic boundary problems with Dirichlet, Neumann and transmission data(c), the globally $H^{1}$ solution is unique, and for compatible outer Neumann data it is unique modulo componentwise constants until the componentwise means are fixed.

(d) [Physics reading and conductor case] on a star-shaped vacuum domain the curl-free $C^{1}$ field has an interior $C^{2}$ potential by Scalar potential and the Poisson equation in a star-shaped electrostatic region. This alone does not supply closure regularity, $H^{1}$ membership, a trace, or prescribed boundary data; each application of (a)-(c) additionally requires its stated regularity and identical boundary/source data. In the bounded conductor model, prescribed absolute conductor potentials $V_{j}$ and absolute outer Dirichlet data determine the potential and field uniquely in the admitted classical or weak class. Grounding fixes the additive constant exactly. If only relative potential differences are prescribed and the outer/reference data are also allowed to shift by the same constant, the common gauge constant is free; merely omitting outer boundary data does not establish field uniqueness.

(e) [Scope] uniqueness is conditional on the stated boundary or decay data: on an unbounded exterior region without a decay condition uniqueness fails: on $Ω_{R} = {x : ∣ x ∣ > R}$ the functions $0$ and $u (x) = 1 - R / ∣ x ∣$ are two distinct bounded harmonic functions with the same zero boundary trace and different limits at infinity, so the boundary, decay and regularity hypotheses above cannot be dropped. No existence is inferred from uniqueness; the classical branch requires $C^{2}$ up to the closure, the weak branches carry their own domain, coefficient and trace hypotheses, and no dielectric material coefficient or empirical claim is included. For the admitted conductor class with connected vacuum, fixed absolute conductor potentials and identical outer Neumann data also give exact uniqueness: this is a mixed boundary problem with a nonempty Dirichlet part, so there is no surviving additive constant. Existence for those mixed data is not asserted here. In that weak mixed branch, write $V_{D} = {v \in H^{1} (Ω_{v}) : T v = 0 on \partial D}$ . The compared potentials have the same prescribed conductor trace and satisfy $ε_{0} \int_{Ω_{v}} \nabla v \cdot \nabla φ = \int_{Ω_{v}} ρ v + ε_{0} \int_{\partial Ω} g T v$ for every $v \in V_{D}$ , with the same smooth outer normal datum $g$ .

Facts & Assumptions

Given: A bounded $C^{\infty}$ domain $Ω_{v}$ with finitely many connected components, a continuous $ρ$ on $\overline{Ω_{v}}$ , classical potentials $φ_{i} \in C^{2} (\overline{Ω_{v}})$ with $- Δ φ_{i} = ρ / ε_{0}$ on $Ω_{v}$ and fields $E_{i} = - \nabla φ_{i}$ , all in the pure electrostatic regime with the SI constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values.

[F1]

Classical Dirichlet and Neumann uniqueness and flux compatibility: for a bounded connected $C^{1}$ domain with $u_{1}, u_{2} \in C^{2} (\overline{Ω})$ , equal $- Δ u_{i}$ and equal Dirichlet traces imply $u_{1} = u_{2}$ on the closure; equal $- Δ u_{i}$ and equal outward normal derivatives imply $u_{1} - u_{2}$ constant; and for a solution $- Δ u = f$ , $\partial_{ν} u = g$ one necessarily has $\int_{Ω} f = - \int_{\partial Ω} g d S$ (Uniqueness of classical Dirichlet and compatible Neumann solutions, Necessary compatibility for the classical Neumann Poisson problem).

[F2]

Componentwise Neumann uniqueness: if a bounded $C^{1}$ domain has finitely many connected components and $u, v \in C^{2} (\overline{Ω})$ solve the same Poisson equation with the same outward normal derivative on $\partial Ω$ , then $u - v$ is constant on each connected component (Classical Neumann solutions differ by componentwise constants).

[F3]

Components of a bounded smooth domain are again bounded smooth domains: each connected component $U$ of $Ω_{v}$ is an open bounded $C^{k}$ domain and its boundary is a union of compact pieces of $\partial Ω_{v}$ (The vacuum region around compact smooth conductors is a bounded smooth domain(d)); adding constants preserves the Poisson equation and the normal derivative (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Classical normal derivative).

[F4]

Weak Dirichlet uniqueness with prescribed trace: for a bounded $C^{1}$ domain and prescribed $g \in H^{1 / 2} (\partial Ω)$ there is exactly one $u \in H^{1} (Ω)$ with trace $g$ solving the weak Dirichlet problem; in particular two weak solutions with the same trace and data coincide (Weak Dirichlet problem with prescribed fractional Sobolev boundary data, The fractional Sobolev space on a compact $C^{1}$ boundary).

[F5]

Weak Neumann existence and uniqueness: a weak Neumann solution with data $ℓ$ exists exactly when $ℓ (1) = 0$ and is then unique up to an additive constant, the normalisation $\int_{Ω} u = 0$ making it unique; on finitely many components both conditions are imposed per component; the transmission problem with sheet charge has the same uniqueness structure (Weak Neumann and transmission existence with interface charges, The fractional Sobolev space on a compact $C^{1}$ boundary).

[F6]

Combined elliptic boundary existence and uniqueness: outer Dirichlet data give a globally $H^{1}$ transmission solution that is unique; compatible outer Neumann data give existence and uniqueness modulo componentwise constants, with zero componentwise means giving uniqueness; the smooth branch is $C^{\infty}$ on each closed side (Weak and classical solvability of the elliptic boundary problems with Dirichlet, Neumann and transmission data(a)-(c)).

[F7]

On a star-shaped open set a curl-free $C^{1}$ field $- E$ has an interior $C^{2}$ potential $φ$ with $\nabla φ = - E$ and $- Δ φ = div E$ ; no boundary regularity or normalisation at infinity is supplied (Scalar potential and the Poisson equation in a star-shaped electrostatic region).

[F8]

The $n = 3$ fundamental solution $Φ (z) = 1 / (4 π ∣ z ∣)$ is smooth and harmonic off its pole, and constant multiples and sums of $C^{2}$ functions with zero Laplacian have zero Laplacian; the radius function satisfies $Δ (1) = 0$ and $Δ (R / ∣ x ∣) = 0$ on $∣ x ∣ > 0$ (The Laplace fundamental solution is harmonic off its pole, The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[F9]

Green's first identity on a bounded $C^{1}$ domain: $\int_{Ω} ∣ \nabla w ∣^{2} = \int_{\partial Ω} w \partial_{ν} w d S$ for harmonic $w \in C^{2} (\overline{Ω})$ with the outward normal $ν$ ; the nonnegative integral of $∣ \nabla w ∣^{2}$ vanishes exactly when $\nabla w = 0$ almost everywhere; a totally differentiable function on a connected open set with vanishing differential is constant (First Green identity, A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere, A differentiable map on a connected open Euclidean set has zero derivative exactly when it is constant).

[F10]

AC and AC $_{ω}$ are declared mathematical assumptions of the cited weak-solvability, trace and measure suppliers (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )); $ε_{0} > 0$ and $Ω_{v}$ , $ρ$ , $φ_{i}$ , $E_{i}$ are the objects of Framework-local SI units, the vacuum constants and the exact/measured status of their values, The static and pure electrostatic regimes and their fixed-time reading and The electric and magnetic fields as fixed-time spatial vector fields; the conductor model supplies the bounded smooth ambient branch and its connectedness requirement (The ideal electrostatic conductor model), and the mean-zero coercivity inequality is available on each connected bounded component (Mean-zero Poincare inequality on a bounded connected $C^{1}$ domain, Euclidean balls have positive finite Lebesgue measure).

Proof

technique · direct

1.1F1F3given

Classical Dirichlet uniqueness. Let $U$ be a connected component of $Ω_{v}$ . By [F3] $U$ is a bounded connected $C^{1}$ domain with $\overline{U} \subseteq \overline{Ω_{v}}$ and the restrictions $φ_{i} ∣_{\overline{U}} \in C^{2} (\overline{U})$ satisfy $- Δ (φ_{i} ∣_{U}) = ρ / ε_{0}$ there, with equal traces on $\partial U$ when $φ_{1} = φ_{2}$ on $\partial Ω_{v}$ . By [F1] claim 1 applied on $U$ , $φ_{1} = φ_{2}$ on $\overline{U}$ ; as $U$ was an arbitrary component, $φ_{1} = φ_{2}$ on all of $\overline{Ω_{v}}$ , and then $E_{1} = - \nabla φ_{1} = - \nabla φ_{2} = E_{2}$ by [F3]. This is (a).

1.2F2F3given

Classical Neumann uniqueness. Assume $\partial_{ν} φ_{1} = \partial_{ν} φ_{2}$ on $\partial Ω_{v}$ . Applying [F2] to the bounded $C^{1}$ domain $Ω_{v}$ with its finitely many components gives that $φ_{1} - φ_{2}$ is constant on each component. Conversely, if $u$ solves the equation and $c_{j}$ is a constant on each component that is added to $u$ , then $Δ (u + c_{j}) = Δ u$ and $\partial_{ν} (u + c_{j}) = \partial_{ν} u$ because the gradient of a constant vanishes [F3], so the equation and the Neumann data are preserved; hence the field $E = - \nabla u$ is unique while the potential is determined only up to componentwise constants. This is the uniqueness half of (b).

1.3F4F5F6given

Weak variants. Clause (c)(i) is the uniqueness content of [F4]: two weak $H^{1}$ Dirichlet solutions with the same prescribed trace and the same right-hand side coincide. Clause (c)(ii) is the uniqueness-modulo-constants and normalisation clause of [F5], with the trace space of The fractional Sobolev space on a compact $C^{1}$ boundary; the componentwise statement is the same clause on each component. Clause (c)(iii) is the uniqueness clause of F6: outer Dirichlet data give a unique globally $H^{1}$ transmission solution, and compatible outer Neumann data give uniqueness modulo componentwise constants, fixed by zero means. This is (c).

1.4F7F10given

Physics reading. On a star-shaped vacuum domain, [F7] supplies an interior $C^{2}$ potential for a curl-free $C^{1}$ field and nothing more: closure regularity, $H^{1}$ membership, a trace and boundary data are additional hypotheses, exactly as assumed in (a)-(c). For the bounded conductor model with its connected vacuum region [F10], fixed absolute conductor potentials and identical outer Dirichlet data are the Dirichlet data of (a) or (c)(i), so the difference of two admitted solutions has zero boundary data and is zero by the same uniqueness clause; grounding fixes the additive constant, while shifting both a relative-potential datum and every reference datum by a common constant leaves the difference of two solutions with the same data and hence with the same (fixed) uniqueness conclusion only for identical data — omitting outer data supplies no field uniqueness. This is (d).

2.1F1F3givenstep 1.2

Neumann flux compatibility. Let $U$ be a connected component of $Ω_{v}$ , with outward normal $ν$ and boundary $\partial U$ a union of compact pieces of $\partial Ω_{v}$ [F3]. Applying [F1] claim 3 (equivalently Necessary compatibility for the classical Neumann Poisson problem) to $u = φ ∣_{U}$ with $f = - Δ φ = ρ / ε_{0}$ and $g = \partial_{ν} φ$ gives $\int_{U} ρ / ε_{0} = - \int_{\partial U} \partial_{ν} φ d S$ , that is $\int_{U} ρ + ε_{0} \int_{\partial U} \partial_{ν} φ d S = 0$ , the displayed identity for the component; no solvability is inferred from it. This completes (b).

2.2F9F10givenstep 1.4

Mixed conductor data. Suppose the vacuum region is connected, $m \geq 1$ , $w$ is the difference of two admitted classical solutions with equal absolute conductor potentials and equal outer Neumann data. Then $w$ is harmonic, $w = 0$ on the nonempty conductor boundary $\partial D$ and $\partial_{ν} w = 0$ on the outer boundary $\partial Ω$ . Green's first identity [F9] gives $\int_{Ω_{v}} ∣ \nabla w ∣^{2} = \int_{\partial Ω_{v}} w \partial_{ν} w d S$ , and each boundary piece carries one zero factor, so the integral vanishes; by [F9], $\nabla w = 0$ almost everywhere, and since $\nabla w$ is continuous on the connected open $Ω_{v}$ it vanishes everywhere, so $w$ is constant by the vanishing-differential characterisation; its zero conductor trace forces the constant to be $0$ . In the weak mixed branch, the difference of two admitted solutions has zero conductor trace, so it is an admissible test in the homogeneous difference equation and the same zero-energy argument applies, with the mean-zero coercivity inequality of [F10] converting $\nabla w = 0$ into $w = w_{Ω_{v}}$ as an $H^{1}$ class whose constant trace on the nonempty conductor boundary is zero. This sharpens (d); existence for mixed data remains unasserted.

3.1F8given∎

The unbounded witness and scope. On $Ω_{R} = {x : ∣ x ∣ > R}$ put $u (x) : = 1 - R / ∣ x ∣$ , and take the comparison potential $φ \equiv 0$ . Then $u \in C^{\infty} (Ω_{R})$ ; since $R / ∣ x ∣ = 4 π R Φ (x)$ is a constant multiple of the $n = 3$ fundamental solution, it is harmonic off the origin [F8], and the constant $1$ has zero Laplacian, so $Δ u = 0$ ; moreover $u = 0$ on $∣ x ∣ = R$ , $0 < u < 1$ on $Ω_{R}$ and $u (x) \to 1$ as $∣ x ∣ \to \infty$ . Thus $0$ and $u$ are two distinct bounded harmonic functions with the same zero boundary trace and different limits at infinity, witnessing that the boundary and decay hypotheses of (a)-(c) cannot be dropped. No existence statement is inferred from uniqueness anywhere above; the classical branch requires $C^{2}$ up to the closure, the weak branches carry their own domain, coefficient and trace hypotheses, and no dielectric coefficient, empirical value or measurement is included. This is (e).

Physics theoremStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Existence and regularity for the electrostatic boundary value problem

Physical scope

Existence and uniqueness of the classical and weak electrostatic boundary value problems on a bounded smooth vacuum region of a named inertial frame: smooth Dirichlet data, compatible smooth Neumann data, and the conductor reading in which constant potentials on compact conductor surfaces together with outer boundary data determine the field and the induced surface charges and total charges. The transmission clause covers finite disjoint smooth interfaces with prescribed sheet charge in vacuum. The statement is conditional on the cited smooth solvability theorem and asserts no existence for prescribed total charges, no nonsmooth geometry, no dielectric material response, no unbounded exterior problem and no empirical claim.

Statement

Assume the Axiom of Choice and Countable Choice as declared by the cited solvability theorem (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $Ω_{v} \subset R^{3}$ be a bounded $C^{\infty}$ domain in the sense of Bounded C^k domains and boundary charts with finitely many connected components (the vacuum region of the additional bounded $C^{\infty}$ ambient branch of The ideal electrostatic conductor model by The vacuum region around compact smooth conductors is a bounded smooth domain, or any bounded smooth domain), let $ρ$ be smooth on $\overline{Ω_{v}}$ (fixed-time volume source density, C m $^{- 3}$ of Framework-local SI units, the vacuum constants and the exact/measured status of their values) and let $ε_{0} > 0$ be the framework constant. Then the electrostatic boundary value problems on $Ω_{v}$ are solvable in the following exact senses, and the solutions are the fields of the pure electrostatic regime The static and pure electrostatic regimes and their fixed-time reading with volume source $ρ$ :

(a) [Dirichlet branch] for every $g \in C^{\infty} (\partial Ω_{v})$ there is exactly one $φ \in C^{\infty} (\overline{Ω_{v}})$ with

$- Δ φ = ρ / ε_{0} on Ω_{v}, φ ∣_{\partial Ω_{v}} = g,$

with $Δ$ the Laplacian of The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field; and it is the unique weak $H^{1}$ solution with trace $g$ ( $C^{k}$ Euclidean maps and diffeomorphisms for the class); the field $E : = - \nabla φ$ (The electric and magnetic fields as fixed-time spatial vector fields) is $C^{\infty}$ on $\overline{Ω_{v}}$ and $(E, 0)$ realises the pure electrostatic regime with source $ρ$ on the vacuum region.

(b) [Neumann branch] for every $g \in C^{\infty} (\partial Ω_{v})$ with

$\int_{Ω_{v}^{(j)}} ρ d x + ε_{0} \int_{\partial Ω_{v}^{(j)}} g d S = 0 for every connected component Ω_{v}^{(j)} of Ω_{v},$

there is $φ \in C^{\infty} (\overline{Ω_{v}})$ with $- Δ φ = ρ / ε_{0}$ and $\partial_{ν} φ = g$ (outward normal $ν$ , Classical normal derivative), unique up to one additive constant per connected component; the normalisation $\int_{Ω_{v}^{(j)}} φ d x = 0$ for every $j$ selects exactly one. The compatibility is also necessary in the sense of Uniqueness and flux compatibility for the electrostatic boundary value problem(b).

(c) [Conductor reading and surface-charge realization] for the separated compact-body conductor geometry and zero-interior-volume convention of The ideal electrostatic conductor model, additionally assuming $Ω$ is a bounded $C^{\infty}$ ambient domain, with outer boundary data $g_{0} \in C^{\infty} (\partial Ω)$ on $\partial Ω$ and constants $V_{j}$ on the conductor surfaces $\partial D_{j}$ (constant functions are smooth on the compact disjoint boundary pieces), the combined datum is a special case of (a); the field $E : = - \nabla φ$ on $Ω_{v}$ extended by $E : = 0$ on the conductor satisfies the adopted distributional vacuum Maxwell equations Distributional vacuum Maxwell equations for the declared point and stationary-sheet sources with the source pair $(ρ_{v} + σ δ_{\partial D}, 0)$ , where $σ : = ε_{0} n \cdot E_{+} \in C^{\infty} (\partial D)$ is the constructed continuous surface-charge density, agreeing afterwards with Surface charge and normal field on an ideal conductor (Distributional Maxwell equations with point and stationary-sheet sources(c), Surface distributions and interfacial jumps, Distributional divergence and curl across a stationary interface); the total charges $Q_{j} = ε_{0} \int_{\partial D_{j}} n \cdot E_{+} d S$ (Surface integration on compact C1 hypersurfaces) are thereby determined by the data, and $φ$ is constant on each $\partial D_{j}$ with value $V_{j}$ .

(d) [Stationary sheet charge in the vacuum region] for finitely many disjoint closed smooth interfaces $Γ$ strictly inside $Ω_{v}$ , a smooth sheet density $σ_{Γ}$ on them, and outer Dirichlet data $g \in C^{\infty} (\partial Ω_{v})$ , there is a weak solution $φ \in H^{1} (Ω_{v})$ which is $C^{\infty}$ on the closure of each side of every $Γ$ , unique by Uniqueness and flux compatibility for the electrostatic boundary value problem(c)(iii), solving the weak equation with right-hand side $\int_{Ω_{v}} ρ v + \sum_{Γ} \int_{Γ} σ_{Γ} T v$ ; its flux jump is $n \cdot [ε_{0} \nabla φ] = - σ_{Γ}$ , equivalently $n \cdot [E] = σ_{Γ} / ε_{0}$ in the sign convention of Stationary vacuum interface jumps and the absence of magnetic sheet sources. This is the vacuum ( $ε = ε_{0}$ ) transmission branch; no dielectric coefficient is introduced.

(e) [Scope] all existence statements are the ones proved in Weak and classical solvability of the elliptic boundary problems with Dirichlet, Neumann and transmission data for its stated smooth bounded domains, smooth uniformly positive coefficients (here $ε = ε_{0} I$ ), smooth data and finitely many disjoint smooth interfaces; the coefficient is the smooth positive constant $ε_{0} I$ on every side in the transmission clause. Excluded: nonsmooth boundaries, corners, edges, coefficient loss of regularity, prescribed total charges for conductor bodies (the free equipotential problem), unbounded exterior regions without a stated decay class, dielectric material response (belonging to the material-response page) and any measurement, apparatus or empirical claim. The theorem asserts existence of the potential, not of a physical equilibrium; the ideal conductor is the adopted model of The ideal electrostatic conductor model.

Facts & Assumptions

Given: A bounded $C^{\infty}$ domain $Ω_{v}$ with finitely many components, smooth $ρ$ on $\overline{Ω_{v}}$ , $ε_{0} > 0$ , and the quoted data of clauses (a)-(d).

[F1]

The combined elliptic boundary existence theorem: for a bounded $C^{\infty}$ domain $Ω$ , smooth uniformly positive coefficient $ε$ and smooth $ρ$ , (a) every smooth Dirichlet datum $g$ admits exactly one weak $H^{1}$ solution with trace $g$ , continuous in the data, $C^{\infty}$ on $\overline{Ω}$ and pointwise satisfying $- div (ε \nabla u) = ρ$ with the given trace; (b) every smooth conormal datum $g$ with $\int_{Ω_{j}} ρ + \int_{\partial Ω_{j}} g = 0$ on every component admits a weak solution, unique up to componentwise constants, normalised by zero componentwise means, smooth on each closed component; (c) the transmission problem with finitely many disjoint closed smooth interfaces strictly inside $Ω$ , piecewise smooth uniformly positive $ε$ , piecewise smooth $ρ$ and smooth sheet charge $σ$ has, with outer Dirichlet data, a unique globally $H^{1}$ solution that is $C^{\infty}$ on each closed side with equal one-sided traces, and with outer Neumann data existence exactly under the componentwise compatibility $\int_{Ω_{j}} ρ + \sum_{Γ \subset Ω_{j}} \int_{Γ} σ + \int_{\partial Ω_{j}} g = 0$ , uniqueness modulo componentwise constants, smoothness on each closed side, and flux jump $n \cdot [ε \nabla u] = - σ$ (Weak and classical solvability of the elliptic boundary problems with Dirichlet, Neumann and transmission data, Classical normal derivative).

[F2]

The pure electrostatic reduction: with $ε = ε_{0} I$ , $- div (ε_{0} \nabla φ) = ρ$ is $- Δ φ = ρ / ε_{0}$ ; $E = - \nabla φ$ then satisfies $div E = ρ / ε_{0}$ and $curl E = 0$ , the fixed-time reduced equations of the regime (The curl of the gradient of a $C^{2}$ function vanishes, The static and pure electrostatic regimes and their fixed-time reading(c), The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, The electric and magnetic fields as fixed-time spatial vector fields, Framework-local SI units, the vacuum constants and the exact/measured status of their values, $C^{k}$ Euclidean maps and diffeomorphisms).

[F3]

The conductor model and its vacuum geometry: in the bounded smooth ambient branch $Ω_{v}$ is a bounded $C^{\infty}$ domain whose boundary pieces are $\partial Ω$ and the conductor surfaces $\partial D_{j}$ , and the model supplies the zero interior field, the connectedness requirement and the potential convention (The ideal electrostatic conductor model, The vacuum region around compact smooth conductors is a bounded smooth domain).

[F4]

Surface-charge identities: for a model of [F3] whose field has the joint one-sided $C^{1}$ regularity, $n \cdot E_{+} = σ / ε_{0}$ , $n \times E_{+} = 0$ , and $Q_{j} = \int_{\partial D_{j}} σ d S = ε_{0} \int_{\partial D_{j}} \partial_{ν} φ d S$ with $ν = - n$ on $\partial D_{j}$ (Surface charge and normal field on an ideal conductor, Surface integration on compact C1 hypersurfaces).

[F5]

Interface differentiation: for a compact embedded $C^{2}$ interface $S$ with unit normal $n$ from $Ω_{-}$ to $Ω_{+}$ and a field $F$ that is $C^{1}$ on each side with one-sided $C^{1}$ traces, the distributional identities $div F = {div F}_{piecewise} + n \cdot [F] δ_{S}$ and $curl F = {curl F}_{piecewise} + n \times [F] δ_{S}$ hold, where the bracket term is the locally integrable function equal to the classical operator on each side (Distributional divergence and curl across a stationary interface, Surface distributions and interfacial jumps).

[F6]

The adopted distributional Maxwell equations and the jump theorem's identities: under ${A C}_{ω}$ , the weak equations of Distributional vacuum Maxwell equations for the declared point and stationary-sheet sources are exactly the solution predicate of Distributional Maxwell equations with point and stationary-sheet sources(e), and the stationary-sheet jump theorem gives $n \cdot [E] = σ / ε_{0}$ , $n \times [E] = 0$ under its one-sided regularity hypotheses (Distributional vacuum Maxwell equations for the declared point and stationary-sheet sources, Distributional Maxwell equations with point and stationary-sheet sources, Stationary vacuum interface jumps and the absence of magnetic sheet sources).

[F7]

Uniqueness and compatibility: the classical and weak uniqueness and compatibility statements of Uniqueness and flux compatibility for the electrostatic boundary value problem hold for bounded smooth vacuum domains with finitely many components, including the componentwise Neumann compatibility and the transmisson uniqueness clause (Uniqueness and flux compatibility for the electrostatic boundary value problem).

[F8]

A bounded smooth domain, in particular the vacuum region $Ω_{v}$ of [F3], has the boundary-chart presentation of Bounded C^k domains and boundary charts used by the solvability theorem [F1], with the finite boundary atlas and outward normals of that definition (Bounded C^k domains and boundary charts, Classical normal derivative).

[F9]

The chain rule differentiates a composition in a boundary chart (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ); a constant function has zero differential (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[F10]

Smooth functions whose derivatives extend continuously to the closure of a bounded smooth domain have a smooth ambient extension, componentwise for vector fields; this extension preserves all boundary derivatives (A fixed smooth-domain extension that preserves smooth parameters).

Proof

technique · direct

1.1F1F2F7F8

Dirichlet branch. By [F8] the bounded smooth domain $Ω_{v}$ carries the boundary-chart presentation required by the solvability theorem [F1]. With $ε : = ε_{0} I$ , smooth and uniformly positive on $\overline{Ω_{v}}$ , and smooth $ρ$ , F1 gives exactly one $u \in H^{1} (Ω_{v})$ with trace $g$ , $C^{\infty}$ on $\overline{Ω_{v}}$ , satisfying $- div (ε_{0} \nabla u) = ρ$ pointwise and $u ∣_{\partial Ω_{v}} = g$ ; the weak uniqueness in the $H^{1}$ class with trace $g$ is part of the same clause. Dividing by $ε_{0} > 0$ gives $- Δ u = ρ / ε_{0}$ and the equation and trace displayed in (a), so $φ : = u$ is the asserted solution and it is unique in the stated class by F7 as well. The field $E : = - \nabla φ$ is $C^{\infty}$ on $\overline{Ω_{v}}$ and satisfies $div E = ρ / ε_{0}$ , $curl E = 0$ by [F2], which is the fixed-time reading of the pure electrostatic regime with source $ρ$ ; hence $(E, 0)$ realises that regime on the vacuum region. This is (a).

1.2F1F2F7

Neumann branch. Let $g \in C^{\infty} (\partial Ω_{v})$ satisfy the componentwise compatibility of (b); with $ε = ε_{0} I$ the conormal datum of F1 is $ε_{0} g$ and its compatibility $\int_{Ω^{(j)}} ρ + \int_{\partial Ω^{(j)}} ε_{0} g = 0$ is exactly the displayed hypothesis, so F1 gives a weak solution, $C^{\infty}$ on each closed component, unique up to one constant per component and normalised by zero componentwise means; after division by $ε_{0}$ the equation is $- Δ φ = ρ / ε_{0}$ and the conormal trace is $\partial_{ν} φ = g$ by the definition of the conormal derivative and the classical normal derivative [F1, F2]. The necessity of the compatibility is F7, whose flux identity gives $\int_{Ω^{(j)}} ρ + ε_{0} \int_{\partial Ω^{(j)}} g d S = 0$ for any classical solution. This is (b).

1.3F1F6F7

Transmission branch. Let the finitely many disjoint closed smooth interfaces $Γ$ , sheet density $σ_{Γ}$ and outer Dirichlet data $g$ be as in (d); with $ε = ε_{0} I$ on each side, piecewise smooth $ρ$ and smooth $σ_{Γ}$ , F1 gives a globally $H^{1}$ weak solution with outer Dirichlet data, $C^{\infty}$ on the closure of each side, equal one-sided traces, unique by F7(iii), and flux jump $n \cdot [ε_{0} \nabla φ] = - σ_{Γ}$ . Since $E = - \nabla φ$ on each side, $[ε_{0} \nabla φ] = ε_{0} [\nabla φ] = - ε_{0} [E]$ , so $n \cdot [E] = σ_{Γ} / ε_{0}$ , the sign convention of [F6]; the weak right-hand side is $\int_{Ω_{v}} ρ v + \sum_{Γ} \int_{Γ} σ_{Γ} T v$ as stated. No dielectric coefficient is introduced because $ε = ε_{0} I$ on every side. This is (d).

2.1F3F4F5F6F9F10step 1.1

Conductor reading. Assume the bounded $C^{\infty}$ ambient branch with outer data $g_{0}$ and conductor data $V_{j}$ . The boundary of $Ω_{v}$ is the disjoint union of the compact boundary pieces $\partial Ω, \partial D_{1}, \dots, \partial D_{m}$ [F3], and the function equal to $g_{0}$ on $\partial Ω$ and to the constant $V_{j}$ on $\partial D_{j}$ is smooth on each piece, hence admissible in (a) [step 1.1]; let $φ$ be the resulting solution and $E = - \nabla φ$ on $Ω_{v}$ , extended by $E : = 0$ on the conductor. Then $φ$ is continuous on $\overline{Ω_{v}}$ with the constant trace $V_{j}$ on each $\partial D_{j}$ [F3], and $E$ has the joint one-sided $C^{1}$ regularity across $S = \partial D$ because $E$ is $C^{\infty}$ on $\overline{Ω_{v}}$ , [F10] supplies its smooth ambient extension, and $0$ is smooth on the conductor side. Define $σ : = ε_{0} n \cdot E_{+} \in C^{\infty} (\partial D)$ directly; its pairing is a stationary-sheet source [F6]. In every boundary chart $X$ , $φ \circ X = V_{j}$ , so the chain rule gives $\nabla φ_{+} \cdot \partial_{a} X = 0$ for both tangent directions, hence $n \times E_{+} = 0$ . Also $ν = - n$ on the conductor boundary, so $\partial_{ν} φ = \nabla φ_{+} \cdot (- n) = n \cdot E_{+}$ and $Q_{j} : = \int_{\partial D_{j}} σ d S = ε_{0} \int_{\partial D_{j}} \partial_{ν} φ d S$ . For the distributional equations: [F5] applied on the compact embedded interface $S$ with $Ω_{-}$ the conductor, $Ω_{+}$ the vacuum and the normal $n$ from conductor to vacuum gives $div E = {div E}_{piecewise} + (n \cdot [E]) δ_{S}$ and $curl E = {curl E}_{piecewise} + (n \times [E]) δ_{S}$ ; the piecewise parts are $ρ_{v} / ε_{0}$ on the vacuum side and $0$ on the conductor side, and $[E] = E_{+}$ with $n \cdot E_{+} = σ / ε_{0}$ , $n \times E_{+} = 0$ by the preceding chart calculation, so the identities are exactly the adopted distributional vacuum Maxwell equations with the source pair $(ρ_{v} + σ δ_{\partial D}, 0)$ [F6]. Extending $φ$ by $V_{j}$ on each body gives the continuous model potential. These verified identities establish the conductor model, so [F4] now applies to it and confirms its surface-charge reading without being used to assume the equations just constructed. This is (c).

3.1F1F3∎

Scope. All existence and regularity statements above are exactly the clauses of [F1] for the smooth bounded domains of the statement, the constant coefficient $ε_{0} I$ , smooth data and finitely many disjoint smooth interfaces; the conductor clause additionally uses only the bounded smooth ambient branch of [F3] and the constant data there. No nonsmooth boundary, corner, edge, coefficient irregularity, prescribed-total-charge problem, unbounded exterior region without a decay class, dielectric response, measurement, apparatus or empirical claim is asserted; existence of the potential is not asserted to be a physical equilibrium, which is the adopted model of The ideal electrostatic conductor model [F3]. This is (e).

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The image-method certificate for a point charge above a grounded plane

Physical scope

The ideal electrostatic problem of a point charge held above an infinite grounded conducting plane, read in the framework's SI conventions: on the vacuum half-space the image potential is harmonic off the source, has the point-charge distributional source, vanishes on the plane, decays like |x|^-2, and is unique in the stated continuous-boundary and decay class; the plane's induced density is the idealization's bookkeeping normal-derivative reading. The item asserts no finite-plane edge effects, no conductor thickness, no dynamics, no self-force and no measurement, and the half-space conductor and noncompact sheet are explicitly outside the bounded-conductor model and the declared source types. The induced density is explicitly smooth on the whole plane, has absolutely convergent total charge $- q$ , and the full-space point-plus-plane divergence identity is verified with local tests as a separate mathematical idealization. Matching singularities allow any remainder of order $o (∣ x - p ∣^{- 1})$ .

Statement

Assume the Axiom of Countable Choice ( ${A C}_{ω}$ ) inherited from the Newtonian, polar and distribution interfaces (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let

$H : = {x \in R^{3} : x_{3} > 0}, Π : = {x_{3} = 0}, p : = a e_{3} (a > 0),$

Here $e_{3} : = (0, 0, 1)$ is the third coordinate unit vector, $a$ carries metres, and the coordinates are SI numerical readings in the named frame. Let $q \in R$ be a charge parameter (coulombs) and $ε_{0} > 0$ the framework constant (Framework-local SI units, the vacuum constants and the exact/measured status of their values). Define, with arbitrary values on the exceptional sets,

$φ_{I} (x) : = \frac{q}{4 π ε_{0}} (\frac{1}{∣ x - p ∣} - \frac{1}{∣ x + a e_{3} ∣}) (x \in H ∖ {p}), E_{I} : = - \nabla φ_{I} .$

The map $E_{I}$ is a spatial field of The electric and magnetic fields as fixed-time spatial vector fields in the fixed-time reading, with the units of Framework-local SI units, the vacuum constants and the exact/measured status of their values. Then:

(a) [Candidate field] $φ_{I} \in C^{\infty} (H ∖ {p})$ and $Δ φ_{I} = 0$ pointwise on $H ∖ {p}$ : the first term is the Coulomb potential of $p$ , whose $n = 3$ fundamental solution is harmonic off its pole (The Laplace fundamental solution is harmonic off its pole, The localized Newtonian potential and the ideal point-charge Coulomb field(d)), and the second term is $C^{\infty}$ and harmonic on the whole of $H$ because its pole $- a e_{3}$ lies outside $H$ ; $E_{I}$ is $C^{\infty}$ on $H ∖ {p}$ .

(b) [Distributional point source] $φ_{I} \in L_{l o c}^{1} (H)$ and

$- Δ T_{φ_{I}} = \frac{q}{ε_{0}} δ_{p} in D^{'} (H),$

where $T_{φ_{I}}$ is the regular distribution of $φ_{I}$ (Regular distribution from a locally integrable function, Locally integrable functions embed in distributions) and $δ_{p}$ the Dirac distribution at $p$ (Dirac delta and its derivatives); moreover the field satisfies $div E_{I} = (q / ε_{0}) δ_{p}$ and $curl E_{I} = 0$ in $D^{'} (H)$ , where the divergence and curl of $E_{I}$ are the distributional derivatives of its components (Distributional derivative). Thus $(E_{I}, 0)$ is a pure electrostatic model on $H$ with the fixed-time point source $q δ_{p}$ , the static reading of the stationary point charge of Distributional Maxwell equations with point and stationary-sheet sources(b) (The static and pure electrostatic regimes and their fixed-time reading(e)).

(c) [Boundary trace and decay] $φ_{I}$ extends continuously to $\overline{H} ∖ {p}$ with $φ_{I} = 0$ on $Π$ (the two denominators are equal there), and for $∣ x ∣ \geq 2 a$ the cancellation bound holds:

$∣ φ_{I} (x) ∣ \leq \frac{2 ∣ q ∣ a}{π ε_{0} ∣ x ∣^{2}} .$

The bound follows by applying the mean value theorem along the segment from $- a e_{3}$ to $a e_{3}$ to the function $y \mapsto ∣ x - y ∣^{- 1}$ , whose gradient is bounded by $(∣ x ∣ - a)^{- 2}$ there (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ , The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Continuity and derivatives of positive-base real powers, For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ). In particular $φ_{I} (x) \to 0$ as $∣ x ∣ \to \infty$ in $H$ .

(d) [Uniqueness in the declared class] let $ψ : H ∖ {p} \to R$ be $C^{2}$ and harmonic on $H ∖ {p}$ , continuous on $\overline{H} ∖ {p}$ , with $ψ = 0$ on $Π$ , $∣ x - p ∣ (ψ (x) - φ_{I} (x)) \to 0$ as $x \to p$ and $ψ (x) \to 0$ as $∣ x ∣ \to \infty$ in $H$ . Then $ψ = φ_{I}$ on $H ∖ {p}$ . Equivalently, the Dirichlet problem $Δ ψ = 0$ on $H ∖ {p}$ , $ψ = 0$ on $Π$ , with the leading singularity of the point charge at $p$ and decay at infinity, has exactly one solution in the stated class.

(e) [Physics reading: ideal grounded plane in the fixed-time reading of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields] the ideal grounded conducting half-space ${x_{3} < 0}$ with the field $E_{I}$ on $H$ , $E \equiv 0$ below $Π$ and the induced density $σ : = ε_{0} n \cdot E_{+} ∣_{Π} = - ε_{0} \partial_{3} φ_{I} ∣_{Π}$ (using the normal $n = e_{3}$ from the conductor into the vacuum; $σ \in C^{\infty} (Π)$ ) is a model of the same equilibrium structure as The ideal electrostatic conductor model: it is a pure electrostatic field on $H$ with the point source at $p$ and the equipotential boundary condition $φ_{I} = 0$ on $Π$ ; the pair $(E_{I}, 0)$ satisfies the static electrostatic equations with source $q δ_{p}$ on $H$ , and the induced density is explicitly $σ (x_{1}, x_{2}) = - q a / [2 π (x_{1}^{2} + x_{2}^{2} + a^{2})^{3 / 2}]$ , is absolutely integrable over the plane, and has total charge $\int_{Π} σ d S = - q$ . However, the half-space conductor is not a compact conductor body: the plane is not a compact embedded hypersurface, $Π$ and ${x_{3} < 0}$ are not among the geometric objects of The ideal electrostatic conductor model, the density $σ$ is not a declared stationary-sheet source of Distributional Maxwell equations with point and stationary-sheet sources(c), and the noncompact plane pairing is not an adopted compact-sheet source of that definition. As a separately specified mathematical idealization, extending $E_{I}$ by zero below the plane gives $div \tilde{E} = (q δ_{p} + σ δ_{Π}) / ε_{0}$ and $curl \tilde{E} = 0$ on $R^{3}$ , where $(σ δ_{Π}) (η) = \int_{R^{2}} σ (x^{'}) η (x^{'}, 0) d x^{'}$ . This verified identity does not extend the framework postulate to arbitrary noncompact sheets. The clause is the explicitly declared idealization whose exact mathematical content is (a)-(d).

(f) [Scope] the statement concerns the ideal point charge above an ideal grounded plane in the declared decay class; it excludes finite-plane edge effects, conductor thickness, transients, moving charges, radiation, self-force and any measurement; no uniqueness is asserted outside the class of (d), and the point-source singularity is never evaluated.

Facts & Assumptions

Given: The half-space $H$ , the plane $Π$ , the point $p = a e_{3}$ , $a > 0$ , the charge parameter $q$ , the constant $ε_{0} > 0$ , and the candidate $φ_{I}$ and field $E_{I}$ of the statement.

[F1]

The $n = 3$ fundamental solution is $Φ (z) = 1 / (4 π ∣ z ∣)$ , smooth and harmonic on $R^{3} ∖ {0}$ ; the Coulomb point-charge branch of The localized Newtonian potential and the ideal point-charge Coulomb field(d) identifies the point-charge potential $φ_{q}^{p} = (q / ε_{0}) Φ (\cdot - p)$ as locally integrable with distributional Laplacian $- Δ T_{φ_{q}^{p}} = (q / ε_{0}) δ_{p}$ on $R^{3}$ and field $E_{q}^{p} = - \nabla φ_{q}^{p}$ satisfying $div E_{q}^{p} = (q / ε_{0}) δ_{p}$ , $curl E_{q}^{p} = 0$ in $D^{'} (R^{3})$ (The Laplace fundamental solution is harmonic off its pole, The localized Newtonian potential and the ideal point-charge Coulomb field, The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, Dirac delta and its derivatives).

[F2]

A $C^{1}$ function on an open set represents its regular distribution and its distributional first derivatives are the regular distributions of its classical derivatives; distributional differentiation is linear and the Laplacian of a classical harmonic function therefore represents the zero distribution (Regular distribution from a locally integrable function, Locally integrable functions embed in distributions, Distributional derivative, Distribution, Local finite order characterization of distributions, If $u, v$ are differentiable on $[a, b]$ with $u^{'}, v^{'}$ integrable, then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$ , Distributional differentiation is continuous and commutes, Test function space d of an open set).

[F3]

Local integrability in dimension three: the radial integral of $∣ z ∣^{- 1}$ over a ball is finite by polar coordinates, finite sums and scalar multiples of locally integrable measurable functions are locally integrable, as are their products with locally bounded measurable functions, and continuous functions are Borel measurable (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma, The polar surface set function on the unit sphere, Integral over a measurable subset, Monotonicity and nonnegative homogeneity of the nonnegative integral, Continuous functions on Euclidean spaces are Borel measurable, Agreement with the existing polar sphere measure, The closed form for the volume of the unit $n$ -ball).

[F4]

Mean value theorem and derivative calculus: for $g (y) : = ∣ x - y ∣^{- 1}$ with $x$ fixed, $g$ is $C^{\infty}$ off $x$ and $\nabla_{y} g (y) = (x - y) / ∣ x - y ∣^{3}$ , so $∣ \nabla_{y} g (y) ∣ = ∣ x - y ∣^{- 2}$ ; differentiation with respect to $x$ instead gives $\nabla_{x} ∣ x - y ∣^{- 1} = - (x - y) / ∣ x - y ∣^{3}$ ; the mean value theorem bounds $∣ g (p) - g (- p) ∣$ by $∣ p - (- p) ∣ \sup ∣ \nabla g ∣$ on the segment (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ , The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Continuity and derivatives of positive-base real powers, For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The finite and reverse triangle inequalities for a norm; and for $n \geq 1$ every norm $N$ on $R^{n}$ satisfies $N (x) \leq C ∥ x ∥_{1}$ and is Lipschitz, hence continuous, for $d_{2}$ , $C^{k}$ Euclidean maps and diffeomorphisms).

[F5]

The half-space uniqueness lemma: a function continuous on $\overline{H} ∖ {p}$ , $C^{2}$ and harmonic on $H ∖ {p}$ , zero on $Π$ , sub-Coulomb at $p$ and vanishing at infinity in $H$ is identically zero (Uniqueness for harmonic functions on a punctured half-space with zero trace, decaying at the pole and at infinity).

[F6]

The polar/Fubini calculus on the plane: integrals over $R^{2}$ may be computed in polar coordinates, the product and Euclidean measures agree on Borel sets, the plane is Lebesgue null in $R^{3}$ , and the one-dimensional fundamental theorem computes the radial primitive (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma, Fubini's theorem for L^1 functions on a sigma-finite product, The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n}, On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n}, A box with a degenerate side is Lebesgue null, and so is every coordinate hyperplane in $R^{n}$ , The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , Riemann–Lebesgue comparison for distribution test integrands, Monotone convergence for the integral).

[F7]

Framework readings: the fixed-time reading of the pure electrostatic regime identifies the static fields and the stationary point charge of Distributional Maxwell equations with point and stationary-sheet sources(b) with $q δ_{p}$ ; the units are those of Framework-local SI units, the vacuum constants and the exact/measured status of their values; and the adopted distributional equations are those of Distributional vacuum Maxwell equations for the declared point and stationary-sheet sources (The static and pure electrostatic regimes and their fixed-time reading(e), The electric and magnetic fields as fixed-time spatial vector fields, Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, Distributional Maxwell equations with point and stationary-sheet sources).

[F8]

For $0 < r < R$ there is a smooth function $χ : R^{3} \to [0, 1]$ equal to one on ${\overline{B}}_{r} (p)$ and supported strictly inside $B_{R} (p)$ (Compactly supported scaled Euclidean bumps).

Proof

technique · direct

1.1F1F2

Clause (a). Write $φ_{I} = (q / ε_{0}) Φ (\cdot - p) - (q / ε_{0}) Φ (\cdot + a e_{3})$ [F1]. The first term is smooth and harmonic off its pole $p$ ; the second has its pole at $- a e_{3}$ , which lies outside the closed half-space $\overline{H}$ , so it is $C^{\infty}$ on $H$ and harmonic there by [F1] and linearity of the Laplacian [F2]; hence $φ_{I} \in C^{\infty} (H ∖ {p})$ , $Δ φ_{I} = 0$ there, and $E_{I} = - \nabla φ_{I}$ is $C^{\infty}$ on $H ∖ {p}$ .

1.2F1F2F3F7

Local integrability and distributional identities. $φ_{I}$ is locally integrable on $H$ : $∣ x - p ∣^{- 1}$ is locally integrable in dimension three and $∣ x + a e_{3} ∣^{- 1}$ is continuous on $H$ , and sums and scalar multiples of locally integrable measurable functions are locally integrable [F3]. Its regular distribution is $T_{φ_{I}} = (q / ε_{0}) T_{Φ (\cdot - p)} - (q / ε_{0}) T_{Φ (\cdot + a e_{3})}$ [F2, F3]. On $H$ the second term is $C^{\infty}$ with classically vanishing Laplacian, so its distributional Laplacian is the regular distribution of the zero function [F2]; the first term has $- Δ T = (q / ε_{0}) δ_{p}$ on $R^{3}$ , hence on the open subset $H$ [F1]. Summing gives $- Δ T_{φ_{I}} = (q / ε_{0}) δ_{p}$ in $D^{'} (H)$ . For the field, $E_{I} = E_{q}^{p} - E_{q}^{- p}$ where $E_{q}^{- p} : = - \nabla ((q / ε_{0}) Φ (\cdot + a e_{3}))$ is smooth on $H$ with $div E_{q}^{- p} = 0$ , $curl E_{q}^{- p} = 0$ there [F2]; componentwise $T_{E_{I, i}} = - \partial_{i} T_{φ_{I}}$ on $H$ by the same representation and [F2], so the distributional divergence and curl of $E_{I}$ are those of the sum, giving $div E_{I} = (q / ε_{0}) δ_{p}$ and $curl E_{I} = 0$ in $D^{'} (H)$ [F1, F2]. This is (b); taking it together with [F7] reads $(E_{I}, 0)$ as the pure electrostatic model on $H$ with the fixed-time point source $q δ_{p}$ , as asserted.

1.3F4

Clause (c). On $Π$ one has $∣ x - p ∣ = ∣ x + a e_{3} ∣$ for every $x$ with $x_{3} = 0$ , so the bracket in $φ_{I}$ vanishes and $φ_{I}$ extends continuously (indeed $C^{\infty}$ away from $x = p \notin Π$ ) with value $0$ on $Π$ . For the decay, fix $x$ with $∣ x ∣ \geq 2 a$ and apply the mean value theorem to $g (y) : = ∣ x - y ∣^{- 1}$ on the segment from $- a e_{3}$ to $a e_{3}$ : since every point $y$ of that segment satisfies $∣ x - y ∣ \geq ∣ x ∣ - a \geq a > 0$ , one has $∣ \nabla g (y) ∣ = ∣ x - y ∣^{- 2} \leq (∣ x ∣ - a)^{- 2}$ and therefore $∣ g (p) - g (- p) ∣ \leq 2 a (∣ x ∣ - a)^{- 2}$ [F4]. Multiplying by $∣ q ∣ / (4 π ε_{0})$ gives $∣ φ_{I} (x) ∣ \leq ∣ q ∣ a / (2 π ε_{0} (∣ x ∣ - a)^{2}) \leq 2 ∣ q ∣ a / (π ε_{0} ∣ x ∣^{2})$ because $∣ x ∣ \leq 2 (∣ x ∣ - a)$ for $∣ x ∣ \geq 2 a$ . This is (c).

2.1F5step 1.1step 1.3

Clause (d). Let $ψ$ be as in the statement and put $w : = ψ - φ_{I}$ on $H ∖ {p}$ . Then $w$ is $C^{2}$ and harmonic on $H ∖ {p}$ and continuous on $\overline{H} ∖ {p}$ by (a) [step 1.1]; $w = 0$ on $Π$ because both functions vanish there [step 1.3]; $∣ x - p ∣ w (x) \to 0$ as $x \to p$ by the assumed leading-singularity condition; and $w (x) \to 0$ as $∣ x ∣ \to \infty$ in $H$ because both $ψ$ and $φ_{I}$ do. By [F5] applied to $w$ , $w = 0$ on $H ∖ {p}$ , so $ψ = φ_{I}$ . This is (d).

2.2F4F6step 1.2

Clause (e), first part. On the plane, $n = e_{3}$ and $σ = - ε_{0} \partial_{3} φ_{I} ∣_{Π}$ ; differentiating the two Coulomb terms, whose gradients are $- (x \mp p) / ∣ x \mp p ∣^{3}$ [F4], at $x_{3} = 0$ gives $\partial_{3} ∣ x - p ∣^{- 1} = a / (x_{1}^{2} + x_{2}^{2} + a^{2})^{3 / 2}$ and $\partial_{3} ∣ x + a e_{3} ∣^{- 1} = - a / (x_{1}^{2} + x_{2}^{2} + a^{2})^{3 / 2}$ , so $\partial_{3} φ_{I} = q a / [2 π ε_{0} (x_{1}^{2} + x_{2}^{2} + a^{2})^{3 / 2}]$ and hence $σ (x_{1}, x_{2}) = - q a / [2 π (x_{1}^{2} + x_{2}^{2} + a^{2})^{3 / 2}]$ , a smooth bounded function; the boundary condition $φ_{I} = 0$ on $Π$ is the equipotential (zero) trace of the ideal grounded plane, and $(E_{I}, 0)$ satisfies the static equations with source $q δ_{p}$ on $H$ by (b) [step 1.2]. The total induced charge is $\int_{R^{2}} σ d λ_{2} = - \frac{q a}{2 π} \int_{0}^{\infty} \frac{2 π r d r}{(r^{2} + a^{2})^{3 / 2}} = - q a [- \frac{1}{\sqrt{r^{2} + a^{2}}}]_{r = 0}^{\infty} = - q,$ with the polar-coordinate computation and the improper radial primitive justified by [F6]; the identical polar computation with $∣ q ∣$ in place of $- q$ gives $\int_{R^{2}} ∣ σ ∣ d λ_{2} = ∣ q ∣ < \infty$ , including $q = 0$ .

3.1F1F2F6F8step 1.2step 1.3step 2.2

Clause (e), extension identity. Extend $E_{I}$ by zero to ${x_{3} \leq 0}$ , obtaining $\tilde{E} \in L_{l o c}^{1} (R^{3})$ by the point-field integrability in [F1] and smoothness away from $p$ . The plane pairing is a distribution: on tests supported in a compact box its absolute value is bounded by $\sup ∣ σ ∣$ times the finite area of the box's planar section times $\sup ∣ η ∣$ . Choose $0 < r < R < a$ and $χ$ from [F8], so $χ = 1$ near $p$ and its support lies in $H$ . Split an arbitrary scalar test as $η_{1} = χ η$ , $η_{2} = (1 - χ) η$ . For $η_{1}$ the half-space identities of step 1.2 give the divergence pairing $(q / ε_{0}) η (p)$ and zero curl pairing; its plane trace is zero. For $η_{2}$ , all field/test products are smooth away from the pole and absolutely integrable on a compact box. Fubini and one-dimensional integration by parts [F2, F6] therefore give, for every $i, j$ , $- \int_{H} E_{I, j} \partial_{i} η_{2} = \int_{H} (\partial_{i} E_{I, j}) η_{2} + 1_{{i = 3}} \int_{R^{2}} E_{I, j} (x^{'}, 0) η_{2} (x^{'}, 0) d x^{'}$ . Horizontal boundary terms vanish by compact support; the vertical integral over $(0, \infty)$ has precisely the displayed lower-end term. Summing $i = j$ gives only $\int_{Π} E_{I, 3} η_{2} = ε_{0}^{- 1} \int_{Π} σ η$ , since $div E_{I} = 0$ away from $p$ . Taking the antisymmetric derivative combinations gives the sheet term $e_{3} \times E_{+} = 0$ , because the smooth zero plane trace of $φ_{I}$ has zero horizontal derivatives; the classical curl also vanishes away from $p$ by step 1.2. Adding the two test pieces gives $div \tilde{E} = (q δ_{p} + σ δ_{Π}) / ε_{0}$ and $curl \tilde{E} = 0$ in $D^{'} (R^{3})$ . No limit at infinity or unproved divergence theorem on a cornered truncation is used.

4.1F7step 2.1step 3.1∎

Scope. The computations above concern exactly the ideal point charge above the ideal grounded plane in the declared class: the plane is not a compact embedded hypersurface and $Π$ , ${x_{3} < 0}$ and the plane pairing are not among the declared conductor or compact stationary-sheet objects of The ideal electrostatic conductor model and Distributional Maxwell equations with point and stationary-sheet sources(c) [F7]; the extension identity of [step 3.1] is a verified statement about the idealization and does not extend the framework postulate to arbitrary noncompact sheets. No finite-plane edge effect, conductor thickness, transient, moving charge, radiation, self-force or measurement is asserted, the singularity at $p$ is never evaluated, and uniqueness is claimed only in the class of (d) [step 2.1]. This is (f).

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Capacitance and energy for specified linear conductor geometry

Physical scope

Capacitance and stored electrostatic energy for a specified linear ideal-conductor geometry in a named inertial frame: with the additional bounded smooth ambient domain, a connected vacuum region, a grounded outer boundary and no volume source, the charge-potential map of the conductor bodies is linear with a symmetric positive definite capacitance matrix, the field energy equals one half of the pairing of potentials and charges, and for a single driven conductor the usual two-terminal relations and positivity of the capacitance hold. The item is conditional on the smooth solvability theorem and asserts no arbitrary-boundary or charge-data solvability, no material response, no transient charging dynamics, no radiation and no empirical claim.

Statement

Assume the Axiom of Choice for the cited unit-Dirichlet solvability and trace interfaces, and its Countable Choice consequence for surface and integral interfaces (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $(Ω; D_{1}, \dots, D_{m}; Ω_{v})$ be the conductor geometry of The ideal electrostatic conductor model, additionally with $Ω$ a bounded $C^{\infty}$ ambient domain and $Ω_{v}$ connected and with the outer boundary $\partial Ω$ grounded: the data on $\partial Ω$ are the constant Dirichlet datum $0$ ; let $ρ_{v} \equiv 0$ on $Ω_{v}$ (the pure capacitor configuration: no volume source), and let the framework constants be those of Framework-local SI units, the vacuum constants and the exact/measured status of their values. For each $j \in {1, \dots, m}$ let $φ_{j} \in C^{\infty} (\overline{Ω_{v}})$ be the unit-potential solution of Existence and regularity for the electrostatic boundary value problem(c) with the smooth data $φ_{j} ∣_{\partial D_{k}} = δ_{j k}, φ_{j} ∣_{\partial Ω} = 0,$ so that $- Δ φ_{j} = 0$ on $Ω_{v}$ with the Laplacian of The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, and let $ν$ be the outward normal of $Ω_{v}$ (so on $\partial D_{j}$ the normal $ν$ points into the conductor). Define the capacitance matrix by $C_{j k} : = ε_{0} \int_{\partial D_{j}} \frac{\partial φ_{k}}{\partial ν} d S, j, k = 1, \dots, m,$ whose entries carry farads (F), with the unit potentials read as numerical volt values; these are flux integrals of continuous functions over the compact embedded hypersurfaces $\partial D_{j}$ (normal derivative of Classical normal derivative) (Surface integration on compact C1 hypersurfaces, The vacuum region around compact smooth conductors is a bounded smooth domain). Then: (a) [Linearity and matrix form] for every $V = (V_{1}, \dots, V_{m}) \in R^{m}$ the unique equilibrium potential with conductor potentials $V$ and grounded outer boundary is $φ^{V} = \sum_{j = 1}^{m} V_{j} φ_{j}$ , and the conductor charges of Surface charge and normal field on an ideal conductor(d) are $Q_{j} (V) = ε_{0} \int_{\partial D_{j}} \frac{\partial φ^{V}}{\partial ν} d S = \sum_{k = 1}^{m} C_{j k} V_{k}, i.e. Q = C V .$ (b) [Reciprocity] $C$ is symmetric: $C_{j k} = C_{k j}$ for all $j, k$ ; the symmetry is Green's symmetric pairing of the harmonic potentials $φ_{j}, φ_{k}$ . (c) [Energy identity] the field energy of the configuration is $U (V) = \frac{ε_{0}}{2} \int_{Ω_{v}} ∣ \nabla φ^{V} ∣^{2} d λ_{3} = \frac{1}{2} V^{T} C V = \frac{1}{2} \sum_{j = 1}^{m} V_{j} Q_{j} (V),$ and $U (V) > 0$ for every $V \neq 0$ : $C$ is symmetric positive definite. In particular $U (V) \geq 0$ , with equality exactly for $V = 0$ , so every nonzero conductor-potential data set stores positive field energy. (d) [One driven terminal] with only conductor $1$ driven and every other conductor grounded, $V = t e_{1}$ , where $e_{1} : = (1, 0, \dots, 0) \in R^{m}$ : $Q_{1} = C_{11} t$ , $Q_{j} = C_{j 1} t$ for $j \neq 1$ , and $U = \frac{1}{2} C_{11} t^{2} = \frac{1}{2} Q_{1} V_{1} = Q_{1}^{2} / (2 C_{11})$ ; the capacitance of the arrangement at conductor $1$ with respect to the grounded reference is the positive geometric number $C_{11} > 0$ , with $C_{11} = Q_{1} / V_{1}$ when $t = V_{1} \neq 0$ , and the energy identities $U = \frac{1}{2} Q V = Q^{2} / (2 C) = \frac{1}{2} C V^{2}$ hold with $C = C_{11}$ , $Q = Q_{1}$ , $V = V_{1}$ . By (c) the positivity holds for every diagonal entry, $C_{j j} > 0$ . (e) [Assembly reading] use the family $V_{j} (s) = s V_{j}$ , $φ_{s} = s φ^{V}$ , $Q_{j} (s) = s Q_{j} (V)$ for $0 \leq s \leq 1$ , with fixed geometry and grounded reference. Prescribe the quasistatic external charge-transfer work by $d W_{e x t} = \sum_{j} V_{j} (s) d Q_{j} (s)$ . Then $W_{e x t} = \int_{0}^{1} s d s \sum_{j} V_{j} Q_{j} (V) = U$ . The potentials vary during assembly; only the final potentials and geometry are fixed. This derives the sheet-charge accounting directly and does not apply the volume-source theorem Finite-source assembly energy equals the electric field energy to singular sheets. The family is an accounting prescription, not an asserted time-dependent Maxwell/material solution. (f) [Scope] the theorem assumes: the compact separated conductor geometry, the additional bounded smooth ambient domain, and connected vacuum region of the model, the grounded outer boundary $\partial Ω$ , zero volume source, and the existence and uniqueness of the unit-potential Dirichlet solutions (Existence and regularity for the electrostatic boundary value problem(c), Uniqueness and flux compatibility for the electrostatic boundary value problem(a)). It asserts no solvability for arbitrary boundary data (the existence theorem is used only for the constant data above), no prescribed-total-charge problem, no dielectric or material response, no finite-conductivity or transient charging dynamics, no radiation, and no measurement; the matrix $C$ is a geometric object of the ideal electrostatic model, and infinite parallel planes require their own explicitly declared idealization, outside this bounded geometry.

Facts & Assumptions

Given: The Axiom of Choice and Countable Choice as declared in the statement; the conductor geometry $(Ω; D_{1}, \dots, D_{m}; Ω_{v})$ of the model in the bounded smooth ambient branch with $Ω$ a bounded $C^{\infty}$ domain, $Ω_{v}$ connected, outer boundary grounded and $ρ_{v} \equiv 0$ ; the unit-potential solutions $φ_{j} \in C^{\infty} (\overline{Ω_{v}})$ with $- Δ φ_{j} = 0$ , $φ_{j} ∣_{\partial D_{k}} = δ_{j k}$ , $φ_{j} ∣_{\partial Ω} = 0$ ; the outward unit normal $ν$ of $Ω_{v}$ ; and the capacitance matrix $C_{j k} = ε_{0} \int_{\partial D_{j}} \partial φ_{k} / \partial ν d S$ .

[F1]

The model and its vacuum geometry: in the bounded smooth ambient branch of The ideal electrostatic conductor model, by The vacuum region around compact smooth conductors is a bounded smooth domain the vacuum region $Ω_{v}$ is a bounded $C^{\infty}$ domain (in particular a bounded $C^{1}$ domain), its boundary is the disjoint union $\partial Ω \cup \partial D_{1} \cup \dots \cup \partial D_{m}$ of nonempty compact embedded smooth hypersurfaces, and $\partial Ω$ is nonempty; the model requires $Ω_{v}$ connected and nonempty, carries the pure electrostatic regime and the fields of the framework, and its constants are those of Framework-local SI units, the vacuum constants and the exact/measured status of their values (The ideal electrostatic conductor model, The vacuum region around compact smooth conductors is a bounded smooth domain, The static and pure electrostatic regimes and their fixed-time reading, The electric and magnetic fields as fixed-time spatial vector fields).

[F2]

Unit-potential solvability and uniqueness: the datum equal to $δ_{j k}$ on $\partial D_{k}$ and $0$ on $\partial Ω$ is constant on each compact boundary piece, hence smooth on the disjoint union $\partial Ω_{v}$ ; with $ρ_{v} \equiv 0$ the conductor reading of Existence and regularity for the electrostatic boundary value problem(c) is a special case of its Dirichlet branch, so there is exactly one $φ_{j} \in C^{\infty} (\overline{Ω_{v}})$ with $- Δ φ_{j} = 0$ on $Ω_{v}$ and the prescribed boundary values; uniqueness in the class of $C^{\infty} (\overline{Ω_{v}})$ functions with identical boundary data is the Dirichlet uniqueness clause (Existence and regularity for the electrostatic boundary value problem(a),(c), Uniqueness and flux compatibility for the electrostatic boundary value problem(a), The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, $C^{k}$ Euclidean maps and diffeomorphisms).

[F3]

Surface charges and fluxes: for an equilibrium solution of the model with the joint one-sided regularity, $Q_{j} = \int_{\partial D_{j}} σ d S = ε_{0} \int_{\partial D_{j}} \partial φ / \partial ν d S$ , where $ν$ is the outward normal of the vacuum; the integrals exist as integrals of the continuous flux over the compact embedded hypersurfaces $\partial D_{j}$ (Surface charge and normal field on an ideal conductor(d), Surface integration on compact C1 hypersurfaces, Classical normal derivative).

[F4]

Green's first identity on the bounded $C^{1}$ domain $Ω_{v}$ : for real $u \in C^{2} (\overline{Ω_{v}})$ and $v \in C^{1} (\overline{Ω_{v}})$ , $\int_{Ω_{v}} (v Δ u + D u \cdot D v) d λ_{3} = \int_{\partial Ω_{v}} v \partial_{ν} u d S$ with the outward normal $ν$ ; the boundary integral splits over the disjoint pieces $\partial Ω, \partial D_{1}, \dots, \partial D_{m}$ (First Green identity, [F1]).

[F5]

Positivity and constancy tools for (c): a continuous function on the compact $\overline{Ω_{v}}$ is bounded and Borel, so for $φ^{V} \in C^{\infty} (\overline{Ω_{v}})$ the function $∣ \nabla φ^{V} ∣^{2}$ is nonnegative, Borel, bounded and integrable over $Ω_{v}$ ; a nonnegative measurable $f$ with $\int f d λ_{3} = 0$ vanishes almost everywhere, and a continuous such $f$ on the connected open $Ω_{v}$ then vanishes everywhere (otherwise a positive value would stay bounded below on a small Euclidean ball of positive finite measure); a totally differentiable function on a nonempty connected open Euclidean set with vanishing differential is constant; a function constant on the connected open $Ω_{v}$ and continuous on $\overline{Ω_{v}}$ has that constant equal to its value $0$ on the nonempty boundary piece $\partial Ω$ (A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere, Euclidean balls have positive finite Lebesgue measure, A differentiable map on a connected open Euclidean set has zero derivative exactly when it is constant, Continuous functions on Euclidean spaces are Borel measurable, Monotonicity and nonnegative homogeneity of the nonnegative integral, Integral over a measurable subset).

[F6]

Path-work reading and the elementary integral: for open $U \subseteq R^{n}$ , $ϕ \in C^{1} (U)$ and a piecewise $C^{1}$ path $γ : [a, b] \to U$ , $\int_{γ} \nabla ϕ \cdot d r = ϕ (γ (b)) - ϕ (γ (a))$ ; and $\int_{0}^{1} s d s = \frac{1}{2}$ by the one-dimensional fundamental theorem (The gradient theorem: the line integral of a gradient is the endpoint increment, The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ).

Proof

technique · direct

1.1F1F2F3given

Setup and well-definedness: by [F1] the vacuum region $Ω_{v}$ is a bounded $C^{1}$ domain (indeed $C^{\infty}$ ), connected, nonempty, with boundary the disjoint union of the nonempty compact pieces $\partial Ω, \partial D_{1}, \dots, \partial D_{m}$ , and with $\partial Ω$ nonempty. By [F2] each unit-potential solution $φ_{j} \in C^{\infty} (\overline{Ω_{v}})$ exists and is unique with $- Δ φ_{j} = 0$ and $φ_{j} ∣_{\partial D_{k}} = δ_{j k}$ , $φ_{j} ∣_{\partial Ω} = 0$ . The normal $ν$ is the outward unit normal of $Ω_{v}$ ; it points into the conductor on each $\partial D_{j}$ [F1]. Each flux $\int_{\partial D_{j}} \partial φ_{k} / \partial ν d S$ exists because $\partial φ_{k} / \partial ν$ is continuous on the compact embedded hypersurface $\partial D_{j}$ carrying finite surface measure [F3], so $C_{j k}$ is well defined.

2.1F2F3givenstep 1.1

Linearity and the charge-potential map (a): fix $V \in R^{m}$ and put $φ^{V} : = \sum_{j = 1}^{m} V_{j} φ_{j} \in C^{\infty} (\overline{Ω_{v}})$ . By linearity of the Laplacian, $- Δ φ^{V} = \sum_{j} V_{j} (- Δ φ_{j}) = 0$ on $Ω_{v}$ ; on $\partial D_{k}$ one has $φ^{V} = \sum_{j} V_{j} δ_{j k} = V_{k}$ , and on $\partial Ω$ , $φ^{V} = 0$ . By the uniqueness clause of [F2], $φ^{V}$ is the unique equilibrium potential with conductor potentials $V$ and grounded outer boundary, and the solution is linear in $V$ by construction. Its charges are given by [F3] and the definition of $C$ : for each $j$ , $Q_{j} (V) = ε_{0} \int_{\partial D_{j}} \partial φ^{V} / \partial ν d S = ε_{0} \sum_{k = 1}^{m} V_{k} \int_{\partial D_{j}} \partial φ_{k} / \partial ν d S = \sum_{k = 1}^{m} C_{j k} V_{k}$ , that is $Q = C V$ .

2.2F4givenstep 1.1

Reciprocity (b): apply Green's first identity [F4] on the bounded $C^{1}$ domain $Ω_{v}$ twice, first with $u = φ_{j}$ , $v = φ_{k}$ and then with $u = φ_{k}$ , $v = φ_{j}$ ; both functions lie in $C^{\infty} (\overline{Ω_{v}})$ and $Δ φ_{j} = Δ φ_{k} = 0$ , so subtracting the two identities gives $0 = \int_{\partial Ω_{v}} (φ_{j} \partial_{ν} φ_{k} - φ_{k} \partial_{ν} φ_{j}) d S$ . The boundary splits into $\partial Ω$ and the conductor surfaces [F4]; on $\partial Ω$ both traces of $φ_{j}, φ_{k}$ vanish, so the outer piece contributes $0$ , while on $\partial D_{ℓ}$ the traces are $δ_{j ℓ}$ and $δ_{k ℓ}$ , so the total reduces to $\int_{\partial D_{j}} \partial_{ν} φ_{k} d S - \int_{\partial D_{k}} \partial_{ν} φ_{j} d S$ . Multiplying by $ε_{0}$ gives $C_{j k} = C_{k j}$ for all $j, k$ ; the symmetry is exactly Green's antisymmetric pairing of the two harmonic potentials.

3.1F3F4F5givenstep 2.1step 2.2

Energy identity and positivity (c): apply Green's first identity [F4] once more with $u = v = φ^{V}$ : $\int_{Ω_{v}} (φ^{V} Δ φ^{V} + ∣ \nabla φ^{V} ∣^{2}) d λ_{3} = \int_{\partial Ω_{v}} φ^{V} \partial_{ν} φ^{V} d S$ . The Laplacian term vanishes by [step 2.1], the outer boundary contributes $0$ because $φ^{V}$ vanishes on $\partial Ω$ , and on $\partial D_{j}$ the trace of $φ^{V}$ is the constant $V_{j}$ , so $\int_{\partial Ω_{v}} φ^{V} \partial_{ν} φ^{V} d S = \sum_{j = 1}^{m} V_{j} \int_{\partial D_{j}} \partial_{ν} φ^{V} d S$ . By [F3] and [step 2.1] this equals $\sum_{j} V_{j} Q_{j} (V) / ε_{0} = \frac{1}{ε_{0}} \sum_{j, k} V_{j} C_{j k} V_{k} = \frac{1}{ε_{0}} V^{T} C V$ ; multiplying by $ε_{0} / 2$ gives the identity $U (V) = \frac{ε_{0}}{2} \int_{Ω_{v}} ∣ \nabla φ^{V} ∣^{2} d λ_{3} = \frac{1}{2} V^{T} C V = \frac{1}{2} \sum_{j} V_{j} Q_{j} (V)$ . The energy is nonnegative because the integrand is; if $U (V) = 0$ , then $\int_{Ω_{v}} ∣ \nabla φ^{V} ∣^{2} d λ_{3} = 0$ , so $∣ \nabla φ^{V} ∣^{2} = 0$ almost everywhere by [F5], hence everywhere by continuity on the connected open $Ω_{v}$ [F5]; thus $φ^{V}$ is constant on $Ω_{v}$ by [F5], and the constant is $0$ because $φ^{V} = 0$ on the nonempty boundary piece $\partial Ω$ and $φ^{V}$ is continuous on the closure [F5]; consequently $V_{j} = φ^{V} ∣_{\partial D_{j}} = 0$ for every $j$ , i.e. $V = 0$ . Hence $U (V) \geq 0$ with equality exactly at $V = 0$ , and since $C$ is symmetric by [step 2.2], the symmetric bilinear form $V \mapsto \frac{1}{2} V^{T} C V$ is positive definite.

4.1givenstep 2.1step 3.1

One driven terminal (d): take $V = t e_{1}$ with $t \in R$ . By [step 2.1], $φ^{V} = t φ_{1}$ , $Q_{1} = C_{11} t$ and $Q_{j} = C_{j 1} t$ for $j \neq 1$ ; by [step 3.1], $U = \frac{1}{2} C_{11} t^{2} = \frac{1}{2} Q_{1} V_{1}$ . For $t = 0$ all charges and the energy vanish while the unit flux defining $C_{11}$ is unchanged; for $t \neq 0$ one may divide by $t = V_{1}$ to obtain $C_{11} = Q_{1} / V_{1}$ and $U = Q_{1}^{2} / (2 C_{11})$ , using $C_{11} > 0$ from the positive definiteness of [step 3.1] (take $V = e_{1}$ : $U (e_{1}) = \frac{1}{2} C_{11} > 0$ ). Hence $C_{11}$ is the positive geometric capacitance of conductor $1$ with respect to the grounded reference, and the displayed identities $U = \frac{1}{2} Q V = Q^{2} / (2 C) = \frac{1}{2} C V^{2}$ hold with $C = C_{11}$ , $Q = Q_{1}$ , $V = V_{1}$ .

4.2F6givenstep 2.1step 3.1

Assembly reading (e): for $0 \leq s \leq 1$ put $V (s) : = s V$ , $φ_{s} : = φ^{s V} = s φ^{V}$ by [step 2.1], and $Q_{j} (s) : = Q_{j} (s V) = s Q_{j} (V)$ , so that $d Q_{j} (s) = Q_{j} (V) d s$ . The prescribed quasistatic external charge-transfer work is $d W_{e x t} = \sum_{j} V_{j} (s) d Q_{j} (s) = \sum_{j} (s V_{j}) Q_{j} (V) d s = s d s \sum_{j} V_{j} Q_{j} (V)$ , and summing over the conductors and integrating over $s \in [0, 1]$ gives $W_{e x t} = (\int_{0}^{1} s d s) \sum_{j} V_{j} Q_{j} (V) = \frac{1}{2} \sum_{j} V_{j} Q_{j} (V) = U$ by [F6] and [step 3.1]. The family is an accounting prescription on the fixed geometry with grounded reference: the potentials vary during assembly, only the final potentials and geometry are fixed, and the sheet-charge accounting is derived directly rather than from the volume-source theorem Finite-source assembly energy equals the electric field energy; the path-work identity of [F6] gives the endpoint potential-drop reading along any interior piecewise $C^{1}$ path, with boundary endpoints understood as limits of the continuous closure traces. This is (e).

5.1F1F2givenstep 2.1step 3.1step 4.1step 4.2∎

Scope (f): the theorem used the compact separated conductor geometry and bounded smooth ambient branch with connected vacuum region [F1], the grounded outer boundary, the zero volume source, and the existence and uniqueness of the unit-potential Dirichlet solutions [F2]; the existence theorem was applied only to the constant data $V$ and $0$ , so no solvability for arbitrary boundary data or for prescribed total charges is asserted, and no dielectric coefficient, material response, finite-conductivity or transient charging dynamics, radiation, measurement or apparatus claim enters [F1, F2]. The matrix $C$ is a geometric object of the ideal electrostatic model of The ideal electrostatic conductor model, and the infinite parallel-plane example is a separate idealization outside this bounded geometry; this completes (f) and the theorem.

5 · Examples, experiments and counterexamples

None yet.

Electrostatic boundaries and conductors

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

The boundary of a bounded smooth domain is a compact embedded hypersurface

Statement

Facts & Assumptions

Proof

The vacuum region around compact smooth conductors is a bounded smooth domain

Statement

Facts & Assumptions

Proof

Uniqueness for harmonic functions on a punctured half-space with zero trace, decaying at the pole and at infinity

Statement

Facts & Assumptions

Proof

The ideal electrostatic conductor model

Postulate

Interpretation and source basis

Surface charge and normal field on an ideal conductor

Statement

Facts & Assumptions

Proof

Uniqueness and flux compatibility for the electrostatic boundary value problem

Statement

Facts & Assumptions

Proof

Existence and regularity for the electrostatic boundary value problem

Statement

Facts & Assumptions

Proof

The image-method certificate for a point charge above a grounded plane

Statement

Facts & Assumptions

Proof

Capacitance and energy for specified linear conductor geometry

Statement

Facts & Assumptions

Proof

5 · Examples, experiments and counterexamples

Sources