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Existence and uniqueness for the weak Dirichlet Poisson problem

Statement

Assume the Axiom of Choice, inherited through the Poincaré supplier named below, together with Countable Choice. Let $Ω \subseteq R^{n}$ be open, nonempty and bounded in one direction, with Poincar'e constant $C_{P}$ for $W_{0}^{1, 2}$ (The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction). For every $F \in H^{- 1} (Ω)$ (The negative Sobolev space $H^{- 1} (Ω)$ ) there is a unique $u \in H_{0}^{1} (Ω)$ with $\int_{Ω} \nabla u \cdot \overline{\nabla v} d x = F (v) for every v \in H_{0}^{1} (Ω),$ that is, the unique weak solution of $- Δ u = F$ with zero boundary values; it satisfies $∥ u ∥_{H_{0}^{1}} \leq (1 + C_{P}^{2}) ∥ F ∥_{H^{- 1}}, and ∥ D u ∥_{L^{2}} \leq (1 + C_{P}^{2})^{1 / 2} ∥ F ∥_{H^{- 1}} .$ Here $- Δ$ is the constant-coefficient operator $L u = - Δ u$ of Uniformly elliptic divergence-form operators and their sesquilinear forms, and the solution is the Lax--Milgram solution for the form $a_{0} (u, v) = \int \nabla u \cdot \overline{\nabla v}$ .

Facts & Assumptions

Given: The Axiom of Choice and Countable Choice; an open, nonempty $Ω \subseteq R^{n}$ bounded in one direction, with Poincar'e constant $C_{P}$ for $W_{0}^{1, 2}$ ; the form $a_{0} (u, v) = \int_{Ω} \nabla u \cdot \overline{\nabla v} d x$ on $H_{0}^{1} (Ω)$ ; and a functional $F \in H^{- 1} (Ω)$ , i.e. a bounded conjugate-linear functional on $H_{0}^{1} (Ω)$ .

[F1]

$H_{0}^{1} (Ω)$ is a Hilbert space for the $W^{1, 2}$ inner product, and $∥ u ∥_{H_{0}^{1}}^{2} = ∥ u ∥_{L^{2}}^{2} + ∥ D u ∥_{L^{2}}^{2}$ (The Sobolev space $H^{1}$ is a Hilbert space, Integer-order Sobolev spaces and their norms, Zero-boundary Sobolev space as a norm closure).

[F2]

$a_{0}$ is the principal form with $a^{i j} = δ^{i j}$ , hence bounded on $H_{0}^{1}$ with bound $M = 1$ and coercive with constant $α = 1 / (1 + C_{P}^{2})$ (The elliptic form is well defined and bounded on $H^{1}$ , Coercivity of the principal Dirichlet form, Uniformly elliptic divergence-form operators and their sesquilinear forms).

[F3]

Poincar'e: $∥ u ∥_{L^{2}} \leq C_{P} ∥ D u ∥_{L^{2}}$ on $H_{0}^{1}$ , so $∥ u ∥_{H_{0}^{1}} \leq (1 + C_{P}^{2})^{1 / 2} ∥ D u ∥_{L^{2}}$ (The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction).

[F4]

Lax--Milgram: bounded coercive forms on a Hilbert space with a bounded conjugate-linear datum have a unique solution, with $α ∥ u ∥ \leq ∥ F ∥$ for the coercivity constant $α$ (The Lax--Milgram theorem, The negative Sobolev space $H^{- 1} (Ω)$ , Weak Dirichlet solutions for a divergence-form operator).

[F5]

Energy identity for any solution: testing with itself gives $Re a_{0} (u, u) = Re F (u)$ and $α ∥ u ∥_{H_{0}^{1}}^{2} \leq Re a_{0} (u, u)$ , so $α ∥ u ∥_{H_{0}^{1}} \leq ∥ F ∥$ (Testing a coercive weak solution with itself gives the energy bound, Complex Lp classes and Euclidean test-function conventions).

Proof

1.1F1F2F4

Existence and uniqueness: by [F2] the form $a_{0}$ is bounded and coercive on the Hilbert space $H_{0}^{1} (Ω)$ ; applying Lax--Milgram [F4] to the bounded conjugate-linear functional $F \in H^{- 1} (Ω)$ gives a unique $u \in H_{0}^{1} (Ω)$ with $a_{0} (u, v) = F (v)$ for every $v \in H_{0}^{1} (Ω)$ , that is, the weak solution of $- Δ u = F$ in the sense of the definition.

1.2F2F5algebra

First estimate: by [F5] applied with $α = 1 / (1 + C_{P}^{2})$ , $∥ u ∥_{H_{0}^{1}} \leq (1 + C_{P}^{2}) ∥ F ∥_{H^{- 1}}$ .

2.1F3F5step 1.2algebra

Gradient estimate: the same substitution read as an identity gives $∥ D u ∥_{L^{2}}^{2} = Re a_{0} (u, u) = Re F (u) \leq ∥ F ∥ ∥ u ∥_{H_{0}^{1}} \leq (1 + C_{P}^{2})^{1 / 2} ∥ F ∥ ∥ D u ∥_{L^{2}}$ by Poincar'e [F3]; dividing by $∥ D u ∥_{L^{2}}$ when it is nonzero (and trivially otherwise) gives $∥ D u ∥_{L^{2}} \leq (1 + C_{P}^{2})^{1 / 2} ∥ F ∥_{H^{- 1}}$ .

3.1step 1.1step 1.2step 2.1∎

Conclusion: for every $F \in H^{- 1} (Ω)$ there is a unique weak solution of the zero-boundary Poisson problem, with the two displayed bounds; the solution is the Lax--Milgram solution for $a_{0}$ .

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Sources

John K. Hunter, Notes on Partial Differential Equations (UC Davis, revised 18 June 2014, complete 242-page two-quarter 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)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (archived 2025 author manuscript) (standard reference, not scraped)
Richard S. Laugesen, Linear Analysis and Partial Differential Equations (University of Illinois, 2020, complete 158-page graduate notes) (standard reference, not scraped)