Parabolic Potential Theory: Basic Facts

Abstract

The potential theory based on the Laplace operator, developed in the preceding chapters, will be called classical potential theory below. The potential theory based on the heat operator $\dot{\Delta }$ and its adjoint $\mathop{\Delta }\limits^{*}$, called parabolic potential theory, will be developed in Chapters XV to XIX. Concepts that are parabolic counterparts of classical concepts will be distinguished by dots or asterisks, depending on whether the concepts are related to $\dot{\Delta }$ or to $\mathop{\Delta }\limits^{*}$. Just as the domains of classical potential theory are subsets of ℝ^N, the domains of parabolic potential theory are subsets of “space time” ${{\mathbb{R}}^{{N + 1}}}$, which we denote in this context by ${{\dot{\mathbb{R}}}^{N}}$. Here $N \geqslant 1$, and the case N = 1 is not exceptional. A point $\dot{\xi } = (\xi ,s)$ of ${{\dot{\mathbb{R}}}^{N}}$ has space coordinate ξ in ℝ^N and time coordinate s = ord $\dot{\xi }$ (the ordinate of $\dot{\xi }$), a point of ℝ. The point $\dot{\eta }:(\eta ,t)$ will be said to be [strictly] below $\dot{\xi }:(\xi ,s)$ if $t \leqslant s\left[ {t < s} \right]$. If $\dot{\xi }$ is a point of an open subset $\dot{D}$ of ${{\dot{\mathbb{R}}}^{N}}$, the set of points of $\dot{D}$ [strictly] below $\dot{\xi }$ relative to $\dot{D}$ is the set of points of $\dot{D}$ that are endpoints of continuous [strictly] downward-directed arcs from $\dot{\xi }$. That is, $\dot{\eta }$ is [strictly] below $\dot{\xi }$ relative to $\dot{D}$ if and only if there is a continuous function f from [0,1] into $\dot{D}$ for which f(0) = $\dot{\xi }$,/(1) = $\dot{\eta }$, and ord f is a [strictly] decreasing function. The upper [lower] half-space of ${{\dot{\mathbb{R}}}^{N}}$ is the set $\left\{ {{\text{ord}}\dot{\xi } > 0} \right\}{\text{ }}\left[ {\left\{ {{\text{ord}}\dot{\xi } < 0} \right\}} \right]$ and the abscissa hyperplane is the set $\left\{ {{\text{ord}}\dot{\xi } = 0} \right\}$. The boundary of a subset of ${{\dot{\mathbb{R}}}^{N}}$ relative to the one-point compactification of ${{\dot{\mathbb{R}}}^{N}}$ will be called the Euclidean boundary, and boundary will mean this boundary unless a different one is specified.