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Neumann Poisson data require a flux compatibility equation

Statement refuted

False claim: Let $n \geq 2$ and let $Ω \subset R^{n}$ be a bounded $C^{1}$ domain. Every pair of smooth data, a smooth $f$ on $\overline{Ω}$ and a smooth $g$ on $\partial Ω$ , admits a solution $u \in C^{2} (\overline{Ω})$ of the classical Neumann problem $- Δ u = f in Ω, \partial_{ν} u = g on \partial Ω,$ with no compatibility condition relating $f$ and $g$ .

The claim fails already at the level of a necessary equation: every such solution must satisfy $\int_{Ω} f d x = - \int_{\partial Ω} g d S,$ and the smooth constant data $f \equiv 1$ , $g \equiv 0$ violate it, because the left-hand side is $λ (Ω) > 0$ while the right-hand side is $0$ . The counterexample below proves both facts. Compatibility is thus necessary, not sufficient: this item asserts no existence result for compatible data.

Facts & Assumptions

Given: Countable Choice, an integer $n \geq 2$ , a bounded $C^{1}$ domain $Ω \subset R^{n}$ in the convention of Bounded C1 domains and their outward normals, and the constant data $f \equiv 1$ on $Ω$ , $g \equiv 0$ on $\partial Ω$ . Write $λ$ for Lebesgue measure on $R^{n}$ .

[A1]

Countable Choice, written ${A C}_{ω}$ , says that every sequence of nonempty sets has a choice function (The Axiom of Countable Choice ( ${A C}_{ω}$ )). It is the only choice principle used below, through the divergence theorem and the Lebesgue-measure interfaces.

[F1]

For $n \geq 2$ , a bounded $C^{1}$ domain $Ω$ and $F \in C^{1} (\overline{Ω}; R^{n})$ , $\int_{Ω} div F d x = \int_{\partial Ω} F \cdot ν d S$ , both integrals finite, with the outward normal on every boundary component (Divergence on a bounded C1 Euclidean domain).

[F2]

The Laplacian is $Δ u = div \nabla u$ for $u \in C^{2} (\overline{Ω})$ , and the classical normal derivative is $\partial_{ν} u = D u \cdot ν$ for the outward unit normal (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, Classical normal derivative).

[F3]

A bounded $C^{1}$ domain is a nonempty bounded open subset of $R^{n}$ with $n \geq 2$ and locally $C^{1}$ boundary; for $C^{2} (\overline{Ω})$ the derivatives through order two extend continuously to $\overline{Ω}$ , and $F \in C^{1} (\overline{Ω}; R^{n})$ means $F$ and its first derivatives extend continuously (Bounded C1 domains and their outward normals).

[F4]

A subset $U$ of a metric space is open exactly when every $x \in U$ has a real $r > 0$ with $B (x, r) \subseteq U$ (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); here $B (a, r) = {y : ∥ y - a ∥_{2} < r}$ is the Euclidean metric ball (Open ball, closed ball and sphere in a metric space).

[F5]

Every Euclidean ball $B (a, r)$ with $r > 0$ is Lebesgue measurable and satisfies $0 < λ (B (a, r)) < \infty$ , under Countable Choice (Euclidean balls have positive finite Lebesgue measure).

[F6]

The Borel $σ$ -algebra is generated by the open sets (The Borel sigma-algebra of a topological space), and under Countable Choice every Borel subset of $R^{n}$ is Lebesgue measurable; in particular every open set is (Assuming countable choice, every Borel subset of $R^{n}$ is Lebesgue measurable).

[F7]

For a measurable set $E$ the integral over $E$ is $\int_{E} h d λ = \int h χ_{E} d λ$ (Integral over a measurable subset); the simple integral of $χ_{E} = 1 \cdot χ_{E}$ is $1 \cdot λ (E)$ (The integral of a nonnegative simple function), and the nonnegative integral of a nonnegative simple measurable function equals its simple integral (The nonnegative integral agrees with the simple integral on simple functions). Thus $\int_{E} 1 d λ = λ (E)$ for measurable $E$ .

[F8]

If $A \subseteq B$ are measurable subsets of a measure space, then $μ (A) \leq μ (B)$ (Measures are monotone).

[F9]

Surface integration on a compact embedded $C^{1}$ hypersurface is chart integration of the signed integrand through its positive and negative parts, so an integrand that vanishes identically integrates to zero (Surface integration on compact C1 hypersurfaces); the boundary $\partial Ω$ of a bounded $C^{1}$ domain is such a hypersurface with the outward normal of [F3].

[F10]

For $k \in N$ a map is of class $C^{k}$ when all iterated partial derivatives of words of length at most $k$ exist and are continuous ( $C^{k}$ maps and multi-index derivative notation in Euclidean space); a constant map has all positive-order iterated partial derivatives equal to $0$ and is therefore smooth.

[L1]

The refuted claim: every pair of smooth Neumann data on a bounded $C^{1}$ domain admits a $C^{2} (\overline{Ω})$ solution, with no compatibility condition between the source $f$ and the boundary datum $g$ .

Counterexample

technique · direct

1.1givenF3F4

By [F3] the domain $Ω$ is a nonempty bounded open subset of $R^{n}$ with $n \geq 2$ ; fix a point $a \in Ω$ . Applying the openness definition [F4] to $a$ gives $r > 0$ with $B (a, r) \subseteq Ω$ , a Euclidean metric ball in the sense of [F4].

1.2givenF3F10

The source datum $f$ is the constant one on $Ω$ , and the boundary datum $g$ is the identically zero function on $\partial Ω$ , the restriction of the zero constant function on $R^{n}$ . By [F10] a constant map has every positive-order iterated partial derivative equal to $0$ and is of class $C^{k}$ for every $k$ , hence smooth; both data are therefore smooth, and $f$ is constant on the open set $Ω$ .

1.3givenA1F1F2F3

Suppose $u \in C^{2} (\overline{Ω})$ satisfies $- Δ u = f$ in $Ω$ and $\partial_{ν} u = g$ on $\partial Ω$ . By [F3] the gradient field $\nabla u$ lies in $C^{1} (\overline{Ω}; R^{n})$ , so the divergence theorem [F1] applies to it, with Countable Choice [A1]: $\int_{Ω} Δ u d x = \int_{Ω} div \nabla u d x = \int_{\partial Ω} \nabla u \cdot ν d S = \int_{\partial Ω} g d S$ , the middle identity by [F1], the first by [F2], and the last because $\nabla u \cdot ν = \partial_{ν} u = g$ by [F2]. Since $f = - Δ u$ pointwise, this says $\int_{Ω} f d x = - \int_{\partial Ω} g d S$ : every classical solution forces the compatibility equation.

2.1step 1.1step 1.2F5F6F7F8

For the constant one datum of step 1.2, $\int_{Ω} f d x = \int_{Ω} 1 d λ = λ (Ω)$ : by [F7] the integral over the measurable set $Ω$ of the constant one is the simple integral of the indicator $χ_{Ω}$ , namely $1 \cdot λ (Ω)$ . Since $Ω$ is open it is Lebesgue measurable by [F6], and $B (a, r) \subseteq Ω$ by step 1.1 with both sets measurable, so the monotonicity [F8] gives $λ (Ω) \geq λ (B (a, r))$ , while [F5] gives $λ (B (a, r)) > 0$ . Hence $\int_{Ω} f d x > 0$ .

2.2step 1.2F3F9

For the zero datum of step 1.2 the boundary integrand $y \mapsto g (y)$ is identically zero on the compact $C^{1}$ hypersurface $\partial Ω$ , so by [F9] its surface integral vanishes: $\int_{\partial Ω} g d S = 0$ , and therefore $- \int_{\partial Ω} g d S = 0$ .

3.1step 1.3step 2.1step 2.2L1algebracases∎

If a solution $u \in C^{2} (\overline{Ω})$ existed, step 1.3 would give $\int_{Ω} f d x = - \int_{\partial Ω} g d S$ , whereas step 2.1 gives $\int_{Ω} f d x > 0$ and step 2.2 gives $- \int_{\partial Ω} g d S = 0$ ; the resulting $0 < \int_{Ω} f d x = 0$ is a contradiction. Hence the smooth constant data $f \equiv 1$ , $g \equiv 0$ admit no classical solution on any bounded $C^{1}$ domain, although by step 1.2 each datum is smooth, and the claim [L1] is refuted. The compatibility equation of step 1.3 is necessary only; its sufficiency for existence is not asserted, and the nonempty domain, positive radius and both boundary orientations are the ones fixed in [F3], [F4] and [F9]. The only choice principle used is ${A C}_{ω}$ of [A1], through the divergence and measure interfaces.

Source notes

Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript), §5.4 equation (5.43), printed p.128, records the Neumann compatibility identity in the present sign convention $- Δ u = f$ . Hunter, Notes on Partial Differential Equations (2014), §2.5 Theorem 2.24 and the surrounding Green identities, printed p.32, supplies the divergence-theorem derivation of the same identity. Neither source is used as a proof here: the necessity is derived directly from the local divergence theorem, and the violation by the constant data $f \equiv 1$ , $g \equiv 0$ is computed from the positivity of the Lebesgue measure of the nonempty open domain. The sibling corollary in this pair (draft at the time of writing) records the same compatibility equation; the derivation here is self-contained and also covers domains that are not connected, which is why no connectedness hypothesis appears. This item asserts no existence theorem for compatible data.

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Sources

Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript) (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (2014) (standard reference, not scraped)