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Weak parabolic maximum principle

Statement

Let $Q = Ω \times (0, T]$ be a parabolic cylinder (Parabolic cylinder and parabolic boundary) with $Ω$ bounded, and let $u \in C^{2, 1} (\overline{Q})$ satisfy $u_{t} - Δ u \leq 0$ in $Q$ . Then $\max_{\overline{Q}} u = \max_{\partial_{p} Q} u .$

Facts & Assumptions

Given: A parabolic cylinder $Q = Ω \times (0, T]$ with $Ω$ bounded and $u \in C^{2, 1} (\overline{Q})$ with $u_{t} - Δ u \leq 0$ in $Q$ .

[F1]

$\overline{Q}$ is compact, $\partial_{p} Q$ is a closed subset of it, and the class $C^{2, 1} (\overline{Q})$ is the cylinder convention of Parabolic cylinder and parabolic boundary.

[F2]

For every $ε > 0$ the function $v = u + ε ∣ x ∣^{2}$ is a strict subsolution, and the maximum of any strict subsolution is attained on $\partial_{p} Q$ (Strict-subsolution perturbation for the heat operator); the Euclidean squares $∣ x ∣^{2}$ are those of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ .

[F3]

Continuous functions on the nonempty compact sets $\overline{Q}$ and $\partial_{p} Q$ attain their maxima (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value, 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, Open cover, subcover, compact metric space, and compact subset of a metric space), and the Archimedean property lets a quantity bounded by $C + ε R^{2}$ for every $ε > 0$ be bounded by $C$ (Every complete ordered field is Archimedean).

Proof

Given: A bounded parabolic cylinder $Q$ and $u \in C^{2, 1} (\overline{Q})$ with $u_{t} - Δ u \leq 0$ in $Q$ .

1.1F1F3given

By [F1] and [F3] the maxima $\max_{\overline{Q}} u$ and $\max_{\partial_{p} Q} u$ exist, and $R^{2} : = \sup_{x \in \overline{Ω}} ∣ x ∣^{2}$ is finite because $Ω$ is bounded; also $u \leq \max_{\overline{Q}} u$ and, for every $x \in \overline{Ω}$ , $∣ x ∣^{2} \leq R^{2}$ .

2.1step 1.1F2F3given

For every $ε > 0$ put $v : = u + ε ∣ x ∣^{2}$ ; by [F2] $v$ is a strict subsolution, so $\max_{\overline{Q}} v = \max_{\partial_{p} Q} v$ , and $\max_{\overline{Q}} u \leq \max_{\overline{Q}} v = \max_{\partial_{p} Q} v \leq \max_{\partial_{p} Q} u + ε R^{2}$ by step 1.1; letting $ε ↓ 0$ and using the Archimedean property [F3] gives $\max_{\overline{Q}} u \leq \max_{\partial_{p} Q} u$ .

3.1step 2.1given∎

Since $\partial_{p} Q \subseteq \overline{Q}$ , the reverse inequality $\max_{\partial_{p} Q} u \leq \max_{\overline{Q}} u$ is immediate, so the two maxima coincide and the weak maximum principle holds.

Depends on

Used by

Comparison and uniqueness for the bounded-cylinder heat problem Corollary
The final-time face is not part of the parabolic boundary Counterexample
Maximum principle on the whole space under Gaussian growth Lemma

Dependency tree · two levels

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Sources

Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (Universitext, Springer 2011) (standard reference, not scraped)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, AMS Graduate Studies in Mathematics) (standard reference, not scraped)
Victor Ivrii, Partial Differential Equations (University of Toronto, 2018, CC BY-SA) (standard reference, not scraped)