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The Neumann Poisson problem is not coercive on all of $H^{1}$

Statement refuted

Assume the Axiom of Choice inherited through the cited suppliers, together with Countable Choice. Let $Ω \subseteq R^{n}$ be a nonempty bounded open set and consider the form $a (u, v) = \int_{Ω} \nabla u \cdot \overline{\nabla v} d x$ on $H^{1} (Ω)$ with the inner product of The Sobolev space $H^{1}$ is a Hilbert space. The constant function $1$ satisfies $a (1, 1) = 0$ while $∥ 1 ∥_{H^{1}} = ∣ Ω ∣^{1 / 2} > 0$ , so no $α > 0$ can satisfy $Re a (u, u) \geq α ∥ u ∥_{H^{1}}^{2}$ for all $u \in H^{1} (Ω)$ : the form is not coercive on $H^{1} (Ω)$ , and Lax--Milgram does not apply in that space. The obstruction is exactly the kernel: $a (u, u) = 0$ forces $\nabla u = 0$ , hence $u$ is constant on each connected component (Zero weak gradient gives componentwise constants), and the associated Neumann problem $a (u, v) = F (v)$ for all $v$ can have no solution when $F (1) \neq 0$ while constants give nontrivial solutions of the homogeneous equation. This motivates the mean-zero subspace formulation Weak Neumann solvability on the mean-zero subspace and its compatibility condition.

Facts & Assumptions

Given: The Axiom of Choice and Countable Choice; a nonempty bounded open set $Ω \subseteq R^{n}$ ; the Hilbert space $H^{1} (Ω)$ with inner product $(u, v)_{H^{1}} = (u, v)_{L^{2}} + \sum_{i} (D_{i} u, D_{i} v)_{L^{2}}$ ; the form $a (u, v) = \int_{Ω} \nabla u \cdot \overline{\nabla v} d x$ ; and the constant class $1$ .

[F1]

Coercivity of a sesquilinear form means $Re a (u, u) \geq α ∥ u ∥_{H^{1}}^{2}$ for all $u$ and some $α > 0$ ; boundedness means the same form has a finite bound (Bounded, coercive and symmetric sesquilinear forms).

[F2]

$1 \in H^{1} (Ω)$ with weak gradient $0$ : the classical partial derivatives of the constant are $0$ and are its weak derivatives, and the constant is in $L^{2}$ because $Ω$ has finite measure (Classical derivatives agree with weak derivatives, Euclidean balls have positive finite Lebesgue measure, Integer-order Sobolev spaces and their norms).

[F3]

$∥ 1 ∥_{H^{1}}^{2} = ∥ 1 ∥_{L^{2}}^{2} + ∥ D 1 ∥_{L^{2}}^{2} = ∣ Ω ∣ + 0 > 0$ , since $∣ Ω ∣ > 0$ for a nonempty open set (Euclidean balls have positive finite Lebesgue measure, Integral over a measurable subset, Hilbert space).

[F4]

Zero weak gradient implies componentwise constancy: $a (u, u) = 0$ means $\int_{Ω} ∣ \nabla u ∣^{2} = 0$ , so $\nabla u = 0$ a.e. and $u$ is constant on each connected component of $Ω$ (Zero weak gradient gives componentwise constants, Connected components, quasicomponents, and totally disconnected spaces, Real and imaginary parts, complex conjugation, and modulus).

[F5]

The mean-zero Neumann theorem requires the compatibility $F (1) = 0$ and produces solutions with $\int_{Ω} u = 0$ (Weak Neumann solvability on the mean-zero subspace).

Proof

1.1F2F3

The constant is not infinitesimal for the form: by [F2] the weak gradient of $1$ vanishes, so $a (1, 1) = \int_{Ω} ∣ \nabla 1 ∣^{2} d x = 0$ , while $∥ 1 ∥_{H^{1}}^{2} = ∣ Ω ∣ > 0$ by [F3].

2.1F1step 1.1

Failure of coercivity: if some $α > 0$ satisfied $Re a (u, u) \geq α ∥ u ∥_{H^{1}}^{2}$ for all $u$ , then at $u = 1$ it would give $0 = Re a (1, 1) \geq α ∣ Ω ∣ > 0$ , a contradiction. Hence the form is not coercive on $H^{1} (Ω)$ , and the Lax--Milgram existence theorem does not apply in that space.

3.1F4F5step 2.1∎

The obstruction is the kernel and the compatibility: by [F4], $a (u, u) = 0$ forces $\nabla u = 0$ a.e., so with $u \neq 0$ the constants are nontrivial solutions of the homogeneous equation; the weak equation $a (u, v) = F (v)$ on all of $H^{1} (Ω)$ , tested at $v = 1$ , forces $F (1) = 0$ , so no solution exists when $F (1) \neq 0$ . On a connected extension domain this obstruction is removed by the cited mean-zero formulation and compatibility condition. On a disconnected domain one must remove constants on every component and impose compatibility on each component; global mean zero alone does not remove the kernel.

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Sources

Richard S. Laugesen, Linear Analysis and Partial Differential Equations (University of Illinois, 2020, complete 158-page graduate notes) (standard reference, not scraped)
Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (Springer Universitext, 2011, complete 614-page text) (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (UC Davis, revised 18 June 2014, complete 242-page two-quarter notes) (standard reference, not scraped)

The Neumann Poisson problem is not coercive on all of H1H1