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Comparison and uniqueness for the bounded-cylinder heat problem

Statement

Let $Q = Ω \times (0, T]$ be a parabolic cylinder (Parabolic cylinder and parabolic boundary) with $Ω$ bounded.

(i) If $u, v \in C^{2, 1} (\overline{Q})$ satisfy $u_{t} - Δ u \leq v_{t} - Δ v$ in $Q$ and $u \leq v$ on $\partial_{p} Q$ , then $u \leq v$ on $\overline{Q}$ .

(ii) If $f \in C (Q)$ and $g \in C (\partial_{p} Q)$ are real-valued, then there is at most one $u \in C^{2, 1} (\overline{Q})$ with $u_{t} - Δ u = f$ in $Q$ and $u = g$ on $\partial_{p} Q$ .

Facts & Assumptions

Given: A bounded parabolic cylinder $Q = Ω \times (0, T]$ , functions $u, v \in C^{2, 1} (\overline{Q})$ , and (for part (ii)) data $f \in C (Q)$ and $g \in C (\partial_{p} Q)$ .

[F1]

The cylinder vocabulary and the class $C^{2, 1} (\overline{Q})$ are those of Parabolic cylinder and parabolic boundary.

[F2]

Weak maximum principle: if $w \in C^{2, 1} (\overline{Q})$ satisfies $w_{t} - Δ w \leq 0$ in $Q$ , then $\max_{\overline{Q}} w = \max_{\partial_{p} Q} w$ (Weak parabolic maximum principle).

[F3]

The heat operator is linear on $C^{2, 1} (\overline{Q})$ : differences and sums of solutions are computed pointwise with $Δ = \sum_{i} \partial_{i} \partial_{i}$ (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

Proof

Given: A bounded parabolic cylinder $Q$ , $u, v \in C^{2, 1} (\overline{Q})$ , and data $f, g$ for part (ii).

1.1F1F2F3given

For part (i) put $w : = u - v$ ; by [F3] $w \in C^{2, 1} (\overline{Q})$ and $w_{t} - Δ w = (u_{t} - Δ u) - (v_{t} - Δ v) \leq 0$ in $Q$ , while $w = u - v \leq 0$ on $\partial_{p} Q$ ; [F2] therefore gives $\max_{\overline{Q}} w = \max_{\partial_{p} Q} w \leq 0$ , that is, $u \leq v$ on $\overline{Q}$ .

2.1step 1.1F2F3given

For part (ii) let $u_{1}, u_{2} \in C^{2, 1} (\overline{Q})$ both satisfy $u_{t} - Δ u = f$ in $Q$ and $u = g$ on $\partial_{p} Q$ , and put $w : = u_{1} - u_{2}$ ; by [F3] $w \in C^{2, 1} (\overline{Q})$ with $w_{t} - Δ w = 0$ in $Q$ and $w = 0$ on $\partial_{p} Q$ . Applying part (i), proved in step 1.1, to the pair $(w, 0)$ gives $w \leq 0$ on $\overline{Q}$ , and applying it to the pair $(0, w)$ gives $w \geq 0$ ; hence $w \equiv 0$ and $u_{1} = u_{2}$ , so there is at most one such solution.

3.1step 1.1step 2.1given∎

Steps 1.1 and 2.1 prove the comparison statement (i) and the uniqueness statement (ii) for the bounded-cylinder Dirichlet problem; no sign of the operator beyond the subsolution direction enters, and no additional hypotheses on $\partial Ω$ are used.

Depends on

Used by

Heat comparison preserves an interval of values Example
Supremum norm stability for forced heat problems Theorem

Dependency tree · two levels

27 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

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