CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Uniqueness for the classical dirichlet problem

Statement

Let $n \geq 2$ and let $Ω \subseteq R^{n}$ be bounded, nonempty and open. Two functions $u, v \in C^{2} (Ω) \cap C (\overline{Ω})$ with $Δ u = Δ v$ in $Ω$ and $u = v$ on $\partial Ω$ agree on $\overline{Ω}$ . Thus prescribed classical Poisson equation and Dirichlet data have at most one such solution. Moreover, for equal Laplacians, $\sup_{\overline{Ω}} ∣ u - v ∣ \leq \sup_{\partial Ω} ∣ u - v ∣ .$ In particular, a sequence of such solutions with a common Laplacian and uniformly convergent boundary traces converges uniformly on the closure.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[F1]

On a bounded nonempty open set, $Δ u \geq Δ v$ and boundary $u \leq v$ imply closure $u \leq v$ for $C^{2}$ interior, continuous-closure functions. (Comparison principle for classical subharmonic functions).

Proof

technique · direct

1.1

Comparison applied to $(u, v)$ and then $(v, u)$ gives $u \leq v$ and $v \leq u$ , respectively, proving uniqueness.

F1given

1.2

For possibly different boundary values put $b = \max_{\partial Ω} ∣ u - v ∣$ . It is finite because the boundary is nonempty compact and the difference is continuous. Compare $u$ with $v + b$ and $v$ with $u + b$ ; their Laplacians agree. Hence $∣ u - v ∣ \leq b$ on the closure.

F1algebra

2.1

For a sequence with common Laplacian, the bound just proved applies to every pair of terms. Uniformly convergent boundary traces are uniformly Cauchy, so the solutions are uniformly Cauchy on the closure. Completeness of the real numbers supplies the pointwise limit and the same Cauchy bound makes convergence uniform. The limit is continuous there, as follows by combining a uniform error bound with continuity of a fixed term.

step 1.2algebra∎

Depends on

Comparison principle for classical subharmonic functions

Used by

Smooth sphere data have a harmonic replacement Lemma

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John K. Hunter, Notes on Partial Differential Equations (2014) (standard reference, not scraped)
Tsogtgerel Gantumur, Harmonic functions (2012) (standard reference, not scraped)