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The weak and the classical maximum principles agree on a smooth subsolution

Example

Example. On the unit disc $Ω = B_{1} (0) \subset R^{2}$ consider $L_{0} = - Δ$ (so $a^{i j} = δ^{i j}$ and $b = c = 0$ in Uniformly elliptic divergence-form operators and their sesquilinear forms) and $u (x) = ∣ x ∣^{2} - 1.$ Then $Δ u = 4 \geq 0$ , so $u$ is subharmonic in the classical sense (Subharmonic and superharmonic functions in rn), and $u = 0$ on $\partial Ω$ while $u \leq 0$ in $Ω$ . The classical weak maximum principle for the Laplacian (Weak maximum principle for the laplacian) gives $\sup_{Ω} u = \sup_{\partial Ω} u = 0$ , and the weak maximum principle Weak maximum principle for coercive divergence-form equations gives the same conclusion, because $u$ is also a weak subsolution of $L_{0} u = 0$ in the sense of Weak subsolutions and supersolutions of a divergence-form equation with $\sup_{\partial Ω} u^{+} = 0$ .

Facts & Assumptions

Given: The Axiom of Choice and Countable Choice; the unit disc $Ω = B_{1} (0) \subset R^{2}$ ; the coefficients $a^{i j} = δ^{i j}$ , $b = c = 0$ ; and the function $u (x) = ∣ x ∣^{2} - 1$ .

[F1]

Classical differentiation gives $D_{i} u = 2 x_{i}$ and $Δ u = 4$ on $Ω$ , so $Δ u \geq 0$ : $u$ is subharmonic in the sense of Subharmonic and superharmonic functions in rn, and $u \in C^{2} (\overline{Ω}) \cap H_{0}^{1} (Ω)$ because $u \leq 0$ on $Ω$ with equality exactly on $\partial Ω$ and $u$ is a polynomial (Bounded C^k domains and boundary charts, The kernel of the trace is the closure of the test functions for the zero-trace identification).

[F2]

Classical weak maximum principle for the Laplacian: for a bounded nonempty open $Ω$ and $u \in C^{2} (Ω) \cap C (\overline{Ω})$ with $Δ u \geq 0$ , $\max_{\overline{Ω}} u = \max_{\partial Ω} u$ (Weak maximum principle for the laplacian).

[F3]

Classical-to-weak consistency: if $u \in C^{2} (\overline{Ω}) \cap H_{0}^{1} (Ω)$ and $L u = f$ with $f \in C (\overline{Ω})$ , then $a (u, v) = \int_{Ω} f \overset{ˉ}{v}$ for every $v \in H_{0}^{1} (Ω)$ , for the sesquilinear form $a$ of Uniformly elliptic divergence-form operators and their sesquilinear forms (Classical solutions satisfy the weak formulation).

[F4]

Alternative direct integration by parts: for $v \in H_{0}^{1} (Ω)$ and $u \in C^{2} (\overline{Ω})$ , the Sobolev Gauss-Green formula gives $\int_{Ω} \nabla u \cdot \nabla v d x = - \int_{Ω} v Δ u d x$ , the boundary term vanishing because $T v = 0$ (The Gauss-Green integration-by-parts formula with Sobolev traces, Weak subsolutions and supersolutions of a divergence-form equation).

[F5]

Weak maximum principle for coercive divergence-form equations: under its hypotheses, a weak subsolution $u$ of $L_{0} u = 0$ on a bounded $C^{1}$ domain satisfies ${ess sup}_{Ω} u \leq \sup_{\partial Ω} u^{+}$ (Weak maximum principle for coercive divergence-form equations).

Verification

1.1F1F2

The classical side. By [F1], $u \in C^{2} (Ω) \cap C (\overline{Ω})$ with $Δ u = 4 \geq 0$ and $u = 0$ on $\partial Ω$ , $u \leq 0$ in $Ω$ ; the classical weak maximum principle [F2] therefore gives $\max_{\overline{Ω}} u = \max_{\partial Ω} u = 0$ , and the values $u (r e_{1}) = r^{2} - 1 ↑ 0$ as $r ↑ 1$ give $\sup_{Ω} u = {ess sup}_{Ω} u = 0$ by continuity.

2.1step 1.1F1F3F4

$u$ is a weak subsolution. Since $L_{0} u = - Δ u = - 4$ , [F3] (or, equivalently, the direct integration by parts of [F4]) gives $a_{0} (u, v) = \int_{Ω} (- 4) v d x = - 4 \int_{Ω} v d x \leq 0$ for every nonnegative $v \in H_{0}^{1} (Ω)$ , where $a_{0} (w, v) = \int_{Ω} \nabla w \cdot \nabla v d x$ ; moreover $u \in H_{0}^{1} (Ω)$ with $u \leq 0$ , so $(u - 0)^{+} = 0 \in H_{0}^{1} (Ω)$ and $\sup_{\partial Ω} u^{+} = 0$ in the boundary-order convention of Weak subsolutions and supersolutions of a divergence-form equation. Thus $u$ is a weak subsolution of $L_{0} u = 0$ with zero positive boundary supremum.

3.1step 1.1step 2.1F2F5∎

Agreement of the two principles. Applying [F5] to the weak subsolution of step 2.1 gives ${ess sup}_{Ω} u \leq \sup_{\partial Ω} u^{+} = 0$ , which agrees with the value $0$ computed in step 1.1; the approaching boundary values and continuity in that step supply the reverse inequality, and the example uses only the explicit polynomial, the two maximum principles and the classical-to-weak consistency.

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Sources

Gerald Teschl, Partial Differential Equations: From Classical to Modern (author manuscript, version 11 February 2025; complete 392-page archived text) (standard reference, not scraped)
Armin Schikorra, Partial Differential Equations (University of Pittsburgh, version 4 December 2019; complete 185-page lecture notes) (standard reference, not scraped)
Leon Simon, Lectures on Partial Differential Equations (Stanford University; complete author scan, 118 sheets reproducing the 223 printed pages of the manuscript, two logical pages per sheet) (standard reference, not scraped)