Heat Equation Maximum Principles Duhamel and Smoothing

Let $n\ge 1$, let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be a nonempty bounded open set with closure $\bar{\mathrm{\Omega }}$ and boundary $\mathrm{\partial }\mathrm{\Omega }$ in the sense of Interior, closure, boundary, limit point, isolated point and dense subset of a metric space, and let $T>0$. Write $Q:=\mathrm{\Omega }\times (0,T]$ for the space-time cylinder and $\bar{Q}:=\bar{\mathrm{\Omega }}\times [0,T]\subseteq {\mathbb{R}}^{n+1}$ for its closure in the space-time variable; the final-time face is $\mathrm{\Omega }\times \{T\}$, while the parabolic boundary of $Q$ is ${\mathrm{\partial }}_{p}Q:=(\bar{\mathrm{\Omega }}\times \{0\})\cup (\mathrm{\partial }\mathrm{\Omega }\times [0,T])\mathrm{.}$ No point of the final-time face belongs to ${\mathrm{\partial }}_{p}Q$: the intersections $(\mathrm{\Omega }\times \{T\})\cap (\bar{\mathrm{\Omega }}\times \{0\})$ and $(\mathrm{\Omega }\times \{T\})\cap (\mathrm{\partial }\mathrm{\Omega }\times [0,T])$ are empty because $T>0$ and $\mathrm{\Omega }\cap \mathrm{\partial }\mathrm{\Omega }=\mathrm{\varnothing }$.

A function $u$ on $\bar{Q}$ is of class $C^{2,1}(\bar{Q})$ if it is continuous there, is $C^2$ in $x$ and $C^1$ in $t$ on $Q$, and the functions $u_t$ and $D_x^{\alpha }u$ for $\mathrm{\mid }\alpha \mathrm{\mid }\le 2$ extend continuously to $\bar{Q}$; the multi-index notation and the classes $C^k$ are those of $C^k$ maps and multi-index derivative notation in Euclidean space, and at $t=T$ the time derivative means the left one-sided limit. The relevant operator is ${\mathrm{\partial }}_t-{\mathrm{\Delta }}_x$, the heat operator of The heat operator, the heat equation, and the Cauchy problem, and the phrase "$u_t-\mathrm{\Delta }u\le 0$ in $Q$" always refers to that cylinder, with the one-sided interpretation of $u_t$ on the final-time face.

On this page a classical heat solution means a function with the stated $C^{2,1}$ regularity satisfying the equation pointwise; no second time derivative is required. Formulas that put time first use the canonical coordinate permutation $(x,t)\leftrightarrow (t,x)$; regularity and derivatives always refer to the named spatial and time variables. The analogous interior class has the same continuous derivatives without an up-to-boundary requirement.

Two elementary topological facts are part of the vocabulary. First, $\bar{Q}$ is compact: $\mathrm{\Omega }$ is bounded, so $\bar{\mathrm{\Omega }}$ is closed and bounded in ${\mathbb{R}}^n$, hence compact (Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ 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); then $\bar{Q}=\bar{\mathrm{\Omega }}\times [0,T]$ is closed and bounded in ${\mathbb{R}}^{n+1}$, hence compact by the same theorem (Open cover, subcover, compact metric space, and compact subset of a metric space). Second, ${\mathrm{\partial }}_{p}Q$ is closed in ${\mathbb{R}}^{n+1}$, being the union of the two closed sets $\bar{\mathrm{\Omega }}\times \{0\}$ and $\mathrm{\partial }\mathrm{\Omega }\times [0,T]$; in particular it is a closed subset of $\bar{Q}$ (Interior, closure, boundary, limit point, isolated point and dense subset of a metric space).

• The class $C^{2,1}(\bar{Q})$ is a convention fixing where the one-sided time derivatives live. The cylinder $Q=\mathrm{\Omega }\times (0,T]$ is open in the spatial directions and half-open in time, and equation statements on $Q$ are read at points with $t<T$ in the usual two-sided sense and at $t=T$ with the left derivative. No global smoothness of $\mathrm{\partial }\mathrm{\Omega }$ is assumed; the maximum principles proved later on this page use only this vocabulary.

• Top versus parabolic boundary. The decomposition of $\bar{Q}$ into the parabolic boundary and the cylinder $Q$ is not a topological boundary decomposition: the final-time face is part of the topological boundary of $Q$ but is deliberately excluded from ${\mathrm{\partial }}_{p}Q$, which is exactly the set on which initial and lateral data are prescribed. A later counterexample on the companion page shows that this asymmetry is forced by the sign of the heat operator and cannot be removed.

Let $n\ge 1$, let $U\subseteq {\mathbb{R}}^n$ be open, let $f\in C^2(U)$, and let $a\in U$ be a local maximum of $f$. Then $h^{\mathsf{T}}{H}_f(a)h\le 0\text{for every }h\in {\mathbb{R}}^n;$ in particular $\mathrm{\nabla }f(a)=0$ and $\mathrm{\Delta }f(a)=\mathrm{tr}{H}_f(a)\le 0$.

Let $Q=\mathrm{\Omega }\times (0,T]$ be a parabolic cylinder (Parabolic cylinder and parabolic boundary) with $\mathrm{\Omega }$ bounded, and let $u\in C^{2,1}(\bar{Q})$ satisfy $u_t-\mathrm{\Delta }u\le 0$ in $Q$. Then:

(i) for every $\epsilon >0$ the function $v:=u+\epsilon \mathrm{\mid }x{\mathrm{\mid }}^2$ is in $C^{2,1}(\bar{Q})$ and satisfies $v_t-\mathrm{\Delta }v\le -2n\epsilon <0\text{in }Q;$

(ii) if $w\in C^{2,1}(\bar{Q})$ satisfies $w_t-\mathrm{\Delta }w<0$ in $Q$, then ${\max }_{\bar{Q}}w={\max }_{{\mathrm{\partial }}_{p}Q}w$.

Let $Q=\mathrm{\Omega }\times (0,T]$ be a parabolic cylinder (Parabolic cylinder and parabolic boundary) with $\mathrm{\Omega }$ bounded, and let $u\in C^{2,1}(\bar{Q})$ satisfy $u_t-\mathrm{\Delta }u\le 0$ in $Q$. Then ${\max }_{\bar{Q}}u={\max }_{{\mathrm{\partial }}_{p}Q}u\mathrm{.}$

Assume Countable Choice. Let $n\ge 1$, and let $\mathrm{\Gamma }$ be the heat kernel of The heat kernel on ${\mathbb{R}}^n$ and its causal extension, with $\mathrm{\mid }x-y{\mathrm{\mid }}^2=⟨x-y,x-y⟩$ the Euclidean square of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. For a space-time point $P=(x,t)\in {\mathbb{R}}^n\times \mathbb{R}$ and $r>0$ the heat ball is $E_r(t,x)=E_r(P):=\{(y,s)\in {\mathbb{R}}^{n+1}:s<t,\mathrm{\Gamma }(x-y,t-s)\ge r^{-n}\}\cup \{(x,t)\}\mathrm{.}$ Write $\tau :=t-s$ for the time depth below the top point. Record the following facts.

(i) $E_r(t,x)$ is compact, and $(x,t)$ is its unique point of maximal time;

(ii) for $0<\tau <r^2\mathrm{/}(4\pi )$ the time slice is the closed ball $\{y:(y,t-\tau )\in E_r(t,x)\}=\bar{B}(x,{\rho }_n(\tau )),{\rho }_n(\tau ):=\sqrt{2n\tau \log \frac{r^2}{4\pi \tau }},$ the slice at $\tau =r^2\mathrm{/}(4\pi )$ is the single point $\{x\}$, and the slice is empty for $\tau >r^2\mathrm{/}(4\pi )$; the function ${\rho }_n$ is continuous on $(0,r^2\mathrm{/}4\pi )$ and ${\rho }_n(\tau )\to 0$ as $\tau \downarrow 0$;

(iii) ${\displaystyle \underset{0<\tau <r^2\mathrm{/}4\pi }{\sup }{\rho }_n(\tau )=\frac{r}{2}\sqrt{\frac{2n}{\pi e}}}$, attained at $\tau =r^2\mathrm{/}(4\pi e)$, so $E_r(t,x)\subseteq \bar{B}(x,\frac{r}{2}\sqrt{2n\mathrm{/}\pi e})\times [t-r^2\mathrm{/}4\pi ,t]$;

(iv) the boundary is $\mathrm{\partial }{E}_r(t,x)=\{(y,s):s<t,\mathrm{\Gamma }(x-y,t-s)=r^{-n}\}\cup \{(x,t)\}$, and on the level set $\mathrm{\Gamma }=r^{-n}$ the gradient of $\mathrm{\Gamma }$ is nowhere zero, so that level set is a $C^{\mathrm{\infty }}$ hypersurface.

• The slice formula is algebra. Taking $n$-th roots in $\mathrm{\Gamma }(x-y,\tau )=(4\pi \tau {)}^{-n\mathrm{/}2}{e}^{-\mathrm{\mid }x-y{\mathrm{\mid }}^2\mathrm{/}(4\tau )}\ge r^{-n}$ (The heat kernel on ${\mathbb{R}}^n$ and its causal extension) gives $\mathrm{\mid }x-y{\mathrm{\mid }}^2\le 2n\tau \log \frac{r^2}{4\pi \tau }$, whose right side is ${\rho }_n(\tau {)}^2$; the logarithmic factor is positive exactly for $0<\tau <r^2\mathrm{/}(4\pi )$, vanishes at $\tau =r^2\mathrm{/}(4\pi )$ (forcing $y=x$, so the slice there is the single point $x$), and is negative for $\tau >r^2\mathrm{/}(4\pi )$, where no point of the level set or of its superlevel set remains. This is the only reading of the inequality $\mathrm{\Gamma }\ge r^{-n}$ at the endpoint $\tau =r^2\mathrm{/}(4\pi )$; the enclosed region and its width are unaffected.

• Compactness (i) and the enclosure (iii). First, $E_r(t,x)$ is closed in ${\mathbb{R}}^{n+1}$: if points $(y_k,s_k)$ with $s_k<t$ and $\mathrm{\Gamma }(x-y_k,t-s_k)\ge r^{-n}$ converge to $(y_0,s_0)$ with $s_0<t$, then continuity of $\mathrm{\Gamma }$ gives $\mathrm{\Gamma }(x-y_0,t-s_0)\ge r^{-n}$; if $s_0=t$, then $t-s_k\to 0$ and the slice bound $\mathrm{\mid }x-y_k{\mathrm{\mid }}^2\le 2n(t-s_k)\log \frac{r^2}{4\pi (t-s_k)}\to 0$ forces $y_0=x$, so the limit is the top point (Interior, closure, boundary, limit point, isolated point and dense subset of a metric space). The enclosure in (iii) bounds $E_r(t,x)$; closed and bounded subsets of ${\mathbb{R}}^{n+1}$ are compact (Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ 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). The maximisation of $\tau \log \frac{r^2}{4\pi \tau }$, whose substitution $u=4\pi \tau \mathrm{/}{r}^2$ reduces it to $\frac{r^2}{4\pi }u\log \frac{1}{u}$ with maximum $\frac{r^2}{4\pi e}$ at $u=1\mathrm{/}e$, follows by differentiating $u\log (1\mathrm{/}u)$: its derivative is $\log (1\mathrm{/}u)-1$, positive for $u<1\mathrm{/}e$ and negative for $u>1\mathrm{/}e$ (The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t, Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm). Also $\tau \log (r^2\mathrm{/}(4\pi \tau ))\to 0$ as $\tau \downarrow 0$: writing $\tau =(r^2\mathrm{/}(4\pi ))e^{-v}$ reduces this to a constant times $v{e}^{-v}\to 0$ (The exponential dominates every fixed nonnegative integer power at $+\mathrm{\infty }$). The top point is the unique point of $E_r(t,x)$ with $s=t$, since the defining inequality requires $s<t$.

• The boundary (iv). On the level set $\mathrm{\Gamma }=r^{-n}$ one has $\mathrm{\Gamma }>0$. If the spatial gradient vanishes, ${\mathrm{\nabla }}_y\mathrm{\Gamma }(x-y,\tau )=0$, then $y=x$ by the explicit formula; at such a point the time derivative ${\mathrm{\partial }}_s\mathrm{\Gamma }(x-y,t-s)=-{\mathrm{\partial }}_{\tau }\mathrm{\Gamma }(0,\tau )$ is $-(-\frac{n}{2\tau })\mathrm{\Gamma }(0,\tau )\mathrm{\ne }0$ (Normalisation, parabolic scaling, heat equation and derivative bounds for the heat kernel). Hence the full gradient in $(s,y)$ never vanishes on the level set, which is therefore a $C^{\mathrm{\infty }}$ hypersurface of the open half-space $s<t$ by the implicit function theorem with higher regularity (The parametrized implicit function theorem with $C^k$ regularity), applied with a nonzero gradient component as the dependent coordinate. Nonzero gradient also gives points on both sides of each level point, so the displayed level is the boundary below time $t$. The adjoined top point is a boundary point, since $(x,t-\tau )\in E_r$ for small $\tau >0$ while no point at a time greater than $t$ belongs to $E_r$; it is not isolated.

• Choice. Countable Choice is the declared ambient hypothesis and is carried by the topology and calculus suppliers; the definition and the displayed computations select no sets.

Assume Countable Choice. Let $u$ be of class $C^{2,1}$ on a neighbourhood of the closed heat ball $E:=E_r(t,x)$ of Heat balls and their time slices, and put $f:=u_t-\mathrm{\Delta }u$. Then, with ${\rho }_n$ the slice radius, $u(x,t)={\iint }_E(\mathrm{\Gamma }(x-y,t-s)-r^{-n})f(y,s)dyds+\frac{1}{2r^n}{\int }_{t-r^2\mathrm{/}4\pi }^t\frac{{\rho }_n(t-s)}{t-s}{\int }_{\mathrm{\mid }y-x\mathrm{\mid }={\rho }_n(t-s)}u(y,s)dS(y)ds,$ all integrals absolutely convergent. For $n=1$, the inner sphere integral is the sum over its two points; endpoint slices are irrelevant to the time integral.

Assume Countable Choice. Let $u$ be of class $C^{2,1}$ on a neighbourhood of the closed heat ball $E_r(t,x)$ of Heat balls and their time slices and suppose $u_t-\mathrm{\Delta }u\le 0$ there. Then $u(x,t)\le \frac{1}{2r^n}{\int }_{t-r^2\mathrm{/}4\pi }^t\frac{{\rho }_n(t-s)}{t-s}{\int }_{\mathrm{\mid }y-x\mathrm{\mid }={\rho }_n(t-s)}u(y,s)dS(y)ds\mathrm{.}$ In particular, if $u\le M$ on $E_r(t,x)$ then $u(x,t)\le M$.

Assume Countable Choice. Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open, let $0<t_1<t_0\le T<\mathrm{\infty }$, and let $\gamma :[0,1]\to \mathrm{\Omega }$ be a Lipschitz path with compact image $K$. Then there is $\rho >0$ such that $E_{\rho }(P)\subseteq \mathrm{\Omega }\times (0,T]$ for every $P\in K\times [t_1,t_0]$, and for every $s\in [t_1,t_0)$ there are $N\ge 1$ and points $P_0=(\gamma (0),t_0),P_1,\dots ,P_N=(\gamma (1),s)$, all lying in $K\times [t_1,t_0]$, with $P_{j+1}\in E_{\rho }(P_j)$ for all $0\le j<N$.