Parabolic cylinder and parabolic boundary

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.