Subparabolic, Superparabolic, and Parabolic Functions on a Slab

Abstract

If $\dot{D}$ is the slab ${{\mathbb{R}}^{N}} \times \left] {0,\delta } \right[$, with $0 < \delta + \infty$, the restriction to $\dot{D} \times \dot{D}$ of $\dot{G}$ satisfies the rather vague description of the Green function ${{\dot{G}}_{{\dot{D}}}}$ given in Section XV.7 for smooth regions. It is therefore to be expected from XV(7.3) that the upper boundary of $\dot{D}$ if $\delta < + \infty$ is a parabolic measure null set and that parabolic measure on the lower boundary is given by

$$ {{\dot{u}}_{{\dot{D}}}}(\dot{\xi },d\eta ) = b(s,\xi - \eta ){{l}_{N}}(d\eta ) = \dot{G}(\dot{\xi },(\eta ,0)){{l}_{N}}(dn)\quad \left[ {\dot{\xi } = (\xi ,s)} \right], $$

so that if $\dot{u}$ is parabolic on $\dot{D}$ with boundary function f in some suitable sense on the lower boundary and if $\dot{u}$ is appropriately restricted, then

$$ \dot{u}(\dot{\xi }) = \int_{{{{\mathbb{R}}^{N}}}} {b(s,\xi - \eta )f(\eta ){{l}_{N}}(d\eta )\quad \left[ {\dot{\xi } = (\xi ,s)} \right].} $$

(1)