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An isolated boundary point obstructs pointwise-zero Green data

Statement

Assume Countable Choice and $n \geq 2$ . Let $B = B_{1} (0) = {x \in R^{n} : ∥ x ∥_{2} < 1}$ and $Ω = B ∖ {0}$ . For every pole $y \in Ω$ , there is no harmonic corrector $H_{y} \in C^{2} (Ω) \cap C (\overline{Ω})$ whose boundary values satisfy $H_{y} (z) = Φ (z - y)$ for every $z \in \partial Ω$ . In particular, there is no Dirichlet Green function on $Ω$ in the pointwise-zero-boundary sense of Dirichlet Green function for minus Laplacian. The argument uses the isolated boundary point $0$ and does not rule out weaker potential-theoretic Green kernels.

Facts & Assumptions

Given: Countable Choice, $n \geq 2$ , the normalized kernel $Φ$ of Fundamental solution for the positive operator minus Laplacian, and the Euclidean metric, norm, balls and spheres of $R^{n}$ as the set of functions $n \to R$ , and $d_{1}$ , $d_{2}$ , $d_{\infty}$ are metrics on it, A norm on a real vector space, the induced metric, and the dictionary with the metric axioms, The $p$ -norms $∥ x ∥_{p}$ for rational $p \geq 1$ , and $∥ x ∥_{\infty}$ , Each $∥ \cdot ∥_{p}$ is a norm on $R^{n}$ , and the induced metrics are exactly $d_{1}$ , $d_{2}$ and $d_{\infty}$ of the published metric-spaces page, Open ball, closed ball and sphere in a metric space and Euclidean spheres and closed balls as subspaces of $R^{n}$ .

[A1]

Countable Choice, written ${A C}_{ω}$ , is the exact assumption used by the kernel convention and the named harmonic-replacement, removability and Green-positivity results below (The Axiom of Countable Choice ( ${A C}_{ω}$ )). The explicit Poisson formula defines the ball corrector family; no full Axiom of Choice is used.

[F1]

For the Euclidean norm, $d_{2} (x, z) = ∥ x - z ∥_{2}$ , and the Euclidean sphere is $S_{2} (0, 1) = {z : ∥ z ∥_{2} = 1}$ ; the standard vector $e_{1} = (1, 0, \dots, 0)$ has norm $1$ ( $R^{n}$ 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 $R^{n}$ , Euclidean spheres and closed balls as subspaces of $R^{n}$ ).

[F2]

Every norm satisfies the reverse triangle inequality $∣ ∥ u ∥ - ∥ v ∥ ∣ \leq ∥ u - v ∥$ (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 $d_{2}$ ).

[F3]

Metric balls are open, and $\overline{A}$ and $\partial A$ mean the adherent points and $\overline{A} ∖ int (A)$ , respectively (Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed, Interior, closure, boundary, limit point, isolated point and dense subset of a metric space).

[F14]

For every $r > 0$ there is $N \geq 1$ with $1 / N < r$ (For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ ); consequently $t = 1 / (N + 1)$ satisfies $0 < t < \min {r, 1}$ .

[F4]

A set is bounded when it lies in a metric ball; $B_{1} (0)$ is nonempty and bounded. Every Euclidean ball is convex: for $x, v \in B_{2} (a, r)$ and $t \in [0, 1]$ , the triangle inequality and positive homogeneity of the norm give $∥ (1 - t) x + t v - a ∥ \leq (1 - t) ∥ x - a ∥ + t ∥ v - a ∥ < r$ . Its segment $t \mapsto (1 - t) x + t v$ is a continuous polygonal path in the ball, so the ball is path-connected, and a path-connected space is connected (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, A norm on a real vector space, the induced metric, and the dictionary with the metric axioms, A convex subset of $R^{m}$ contains every line segment between two of its points, A finite concatenation of straight segments in $R^{n}$ is a continuous path, Polygonal paths and polygonally connected subsets of $R^{n}$ , Paths, path-connected spaces and path components, Every path-connected space is connected, and every path component lies inside a component).

[F5]

If $n \geq 2$ and $U \subseteq R^{n}$ is nonempty, open and connected, then $U ∖ {p}$ is nonempty, open, connected and path-connected for every $p \in U$ (Puncturing a connected open subset of $R^{n}$ preserves path-connectedness for $n \geq 2$ ).

[F6]

The unit sphere is a smooth regular level set and has local $C^{\infty}$ graph charts. The level map $F (z) = ⟨ z, z ⟩ = \sum_{i} z_{i}^{2}$ has continuous coordinate partials $\partial_{i} F (z) = 2 z_{i}$ whose further derivatives are constant, so it is $C^{k}$ for every $k$ ; the continuous-partials theorem identifies its total derivative $D F (z) h = 2 ⟨ z, h ⟩$ , and on the unit sphere $D F (z) z = 2 \neq 0$ , so $1$ is a regular value and the regular-level graph theorem supplies local $C^{k}$ graph charts for every $k$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , $C^{k}$ maps and multi-index derivative notation in Euclidean space, $C^{k}$ Euclidean maps and diffeomorphisms, A regular level set is locally a $C^{k}$ graph of dimension $m - n$ , Regular and critical points, regular and critical values, and level sets, Submersions and immersions between Euclidean open sets, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative, 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, 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 $g (c) \neq 0$ ). Smooth Euclidean maps remain smooth under composition ( $C^{k}$ Euclidean maps are closed under componentwise algebra and composition, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[F7]

The normalized kernel is $Φ (x) = ∣ x ∣^{2 - n} / ((n - 2) ω_{n - 1})$ for $n \geq 3$ and $Φ (x) = - (2 π)^{- 1} \log ∣ x ∣$ for $n = 2$ , and its translates are smooth away from their poles (Fundamental solution for the positive operator minus Laplacian, The Laplace fundamental solution is harmonic off its pole).

[F8]

Smooth real data $g$ on $\partial B_{1}$ have a harmonic replacement $h \in C^{\infty} (B_{1}) \cap C ({\overline{B}}_{1})$ equal to $g$ on the sphere; it is given by the displayed Poisson integral, so uniqueness makes the family parameterized by its data (Smooth sphere data have a harmonic replacement under Countable Choice).

[F9]

A Dirichlet Green function is built from correctors $H_{p} \in C^{2} (U) \cap C (\overline{U})$ with $H_{p} (z) = Φ (z - p)$ on $\partial U$ and $G_{U} (x, p) = Φ (x - p) - H_{p} (x)$ (Dirichlet Green function for minus Laplacian).

[F10]

On a bounded, nonempty, open, connected set, any existing Dirichlet Green function satisfies $G_{U} (x, p) > 0$ for distinct $x, p$ (A bounded-domain Dirichlet Green function is unique and positive).

[F11]

A bounded harmonic function on a punctured neighborhood in dimension $n \geq 2$ has a unique harmonic extension across the puncture (Removable singularity for bounded harmonic functions under Countable Choice).

[F12]

If $u \in C^{2} (U) \cap C (\overline{U})$ on a bounded nonempty open set and $Δ u \geq 0$ , then $\max_{\overline{U}} u = \max_{\partial U} u$ (Weak maximum principle for the laplacian).

[F13]

The Laplacian is the sum of pure second derivatives; coordinate derivatives are linear, so the Laplacian of the difference of two harmonic $C^{2}$ functions is zero (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 $g (c) \neq 0$ ).

Counterexample

technique · direct

1.1F1F2F3F4F5F14choosealgebra

By [F1] and [F3], $B_{1} (0)$ is open. It is nonempty since it contains $0$ , and it is bounded because $B_{1} (0) \subset {x : ∥ x ∥_{2} < 2}$ ; it is connected by [F4]. For $∥ z ∥_{2} = 1$ and every $r > 0$ , [F14] gives $N \geq 1$ with $1 / N < r$ ; putting $t = 1 / (N + 1)$ gives $0 < t < \min {r, 1}$ . Then $(1 - t) z \in B_{1} (0)$ and $∥ (1 - t) z - z ∥_{2} = t < r$ , so every ball about $z$ meets $B_{1} (0)$ ; $B_{1} (0)$ is open, hence $z \in \partial B_{1} (0)$ . If $∥ z ∥_{2} > 1$ , then [F2] gives $∥ x ∥_{2} \geq ∥ z ∥_{2} - ∥ x - z ∥_{2} > 1$ whenever $∥ x - z ∥_{2} < ∥ z ∥_{2} - 1$ , so such $z$ lies outside $\overline{B_{1} (0)}$ . Thus $\overline{B_{1} (0)} = {\overline{B}}_{2} (0, 1)$ and $\partial B_{1} (0) = S_{2} (0, 1)$ . The puncturing result [F5] makes $Ω = B_{1} (0) ∖ {0}$ nonempty, open and connected, and it remains bounded as a subset of $B_{1} (0)$ . Every point of $S_{2} (0, 1)$ is adherent to $Ω$ by the same radial approximation, and $0$ is adherent because $t e_{1} \in Ω$ for the positive $t < \min {r, 1}$ supplied by [F14] in every ball of radius $r$ about $0$ . Points of norm greater than $1$ have the disjoint neighborhoods just proved. Since $Ω$ is open, [F3] gives $\overline{Ω} = {\overline{B}}_{2} (0, 1)$ and $\partial Ω = S_{2} (0, 1) \cup {0}$ ; in particular $0$ is an isolated boundary point, with $B_{2} (0, 1 / 2) \cap \partial Ω = {0}$ .

2.1A1F2F6F7F8F9F10step 1.1construct

For each $p \in B_{1} (0)$ , the reverse triangle inequality [F2] gives $∥ z - p ∥_{2} \geq 1 - ∥ p ∥_{2} > 0$ on $S_{2} (0, 1)$ . Thus $g_{p} (z) = Φ (z - p)$ is defined and smooth there: [F7] gives smoothness in an ambient neighborhood of every sphere point, and [F6] gives smoothness after restriction to the sphere's local graph charts. Apply [F8] to obtain the unique harmonic replacement $H_{p}^{B}$ for each $p$ ; its explicit Poisson formula defines this family without a choice function. By [F9] and $\partial B_{1} (0) = S_{2} (0, 1)$ from step 1.1, the function $G_{B} (x, p) : = Φ (x - p) - H_{p}^{B} (x)$ is a Dirichlet Green function on $B_{1} (0)$ . The boundedness, nonemptiness, openness and connectedness required by [F10] were verified in step 1.1, so for every $y \in Ω$ one has $G_{B} (0, y) = Φ (- y) - H_{y}^{B} (0) > 0$ , because $0 \neq y$ .

2.2F9F11F12F13step 1.1algebra

Fix any $y \in Ω$ and suppose a corrector $H_{y}$ on $Ω$ with the stated pointwise boundary values exists. Set $w = H_{y} - H_{y}^{B}$ on $B_{1} (0) ∖ {0}$ . Both terms are $C^{2}$ and harmonic there; [F13] therefore gives that $w$ is harmonic. Step 1.1 identifies $\overline{Ω}$ with the closed unit ball, so the assumed continuous extension of $H_{y}$ and the replacement's continuous extension make $w$ continuous on that closed ball. It is bounded near $0$ by this continuity, and the removable-singularity result [F11] extends it harmonically to $\tilde{w}$ on $B_{1} (0)$ . The extension agrees at $0$ with the continuous trace $w (0)$ , since both are continuous and agree on the punctured ball. On the outer sphere the two correctors have the same boundary data $Φ (z - y)$ , so $\tilde{w} = 0$ there. Apply [F12] to $\tilde{w}$ and $- \tilde{w}$ on $B_{1} (0)$ ; both are harmonic, continuous on the closed ball and zero on its boundary. Hence $\tilde{w} \equiv 0$ .

3.1A1F8F9F10F11step 1.1step 2.1step 2.2algebraassume-contracontradictiondischarge-contradiction∎

But $0 \in \partial Ω$ by step 1.1, so the assumed boundary condition gives $H_{y} (0) = Φ (- y)$ . Consequently $\tilde{w} (0) = w (0) = H_{y} (0) - H_{y}^{B} (0) = G_{B} (0, y) > 0$ by step 2.1, contradicting $\tilde{w} \equiv 0$ . This contradiction holds for every $y \in Ω$ ; the Green definition [F9] requires such a corrector for every pole, so no Green function exists in that pointwise-zero-boundary sense. The proof uses $n \geq 2$ for the puncture and removability results; the pole is always distinct from $0$ , both boundary pieces are treated, and there is no iff assertion. Its only choice assumption is ${A C}_{ω}$ from [A1], the kernel convention and [F8], [F10], [F11]; the ball family itself is given by the explicit formula.

Source notes

Schmidt, Partial Differential Equations I (2026), §2.10 remark (3), printed p.68, lists isolated boundary points among irregular points. The preceding discussion says that the section omits detailed proofs, so this is context only; the obstruction above is proved from the ball replacement, positivity, removability and weak maximum principle with their hypotheses checked. Schmidt's §2.8 Green definition and remarks (0)–(1), printed pp.44–45, use a harmonic corrector and zero boundary values; his kernel convention has the opposite sign, translated here as $Φ = - F$ and $G_{h e r e} = - G_{S c h m i d t}$ . Teschl, §5.4, printed p.126, notes after Lemma 5.22 that a Green representation formula alone does not establish Dirichlet solvability; it is contextual and does not prove this counterexample.

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Sources

Thomas Schmidt, Partial Differential Equations I (2026) (standard reference, not scraped)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript) (standard reference, not scraped)