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

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Weak maximum principle for the laplacian

Statement

Let $n \geq 2$ and let $Ω \subseteq R^{n}$ be bounded, nonempty and open. If $u \in C^{2} (Ω) \cap C (\overline{Ω})$ and $Δ u \geq 0$ , then $\max_{\overline{Ω}} u = \max_{\partial Ω} u .$ No connectedness or boundary smoothness is required.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[F1]

Adding $ε ∣ x ∣^{2}$ to a subharmonic function, with $ε > 0$ , excludes interior local maxima. (Strict subharmonic perturbation).

[F2]

A Euclidean subset is compact if and only if it is closed and bounded. (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[F3]

A continuous real function on a nonempty compact metric space attains its maximum and minimum. (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value).

Proof

technique · direct

1.1

Choose $R > 0$ with $\overline{Ω} \subseteq {\overline{B}}_{R} (0)$ . The closure is nonempty compact. For each $ε > 0$ , continuity makes $u_{ε} = u + ε ∣ x ∣^{2}$ attain its maximum on that closure.

F2F3given

2.1

The maximizer cannot lie in $Ω$ , so it lies in $\partial Ω$ . In particular the boundary is nonempty; it is closed and bounded, hence compact, and $m = \max_{\partial Ω} u$ exists.

F1F2F3step 1.1

3.1

For every $x \in \overline{Ω}$ , $u (x) \leq u_{ε} (x) \leq m + ε R^{2}$ . Letting $ε ↓ 0$ yields $u (x) \leq m$ . Since a boundary maximizer belongs to the closure, equality of the maxima follows.

step 2.1algebra∎

Depends on

Used by

Weak minimum principle for the laplacian Corollary
Weak maximum principle needs boundedness or control at infinity Counterexample
Harmonic function attaining only boundary extrema Example
Maximum principles need domain and boundary hypotheses Remark
Comparison principle for classical subharmonic functions Theorem
Hopf boundary point lemma for the laplacian Theorem
Maximum principle with limsup control at infinity Theorem

Dependency tree · two levels

40 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

Hunter, Notes on Partial Differential Equations (standard reference, not scraped)