The Parabolic Dirichlet Problem, Sweeping, and Exceptional Sets

Abstract

Let $\dot{D}$ be a nonempty open subset of ${{\dot{\mathbb{R}}}^{N}}$, and let $\dot{h}$ be a strictly positive parabolic function on $\dot{D}$. A function $\dot{\upsilon }/\dot{h}$ on $\dot{D}$ will be called $\dot{h}$-parabolic, $\dot{h}$-superparabolic, or $\dot{h}$-subparabolic if $\dot{\upsilon }$ is parabolic, superparabolic, or sub-parabolic, respectively. The notation will be parallel to that in the classical context, with $\dot{h}$ omitted when $\dot{h} \equiv 1$. Thus $\dot{G}M_{{\dot{D}}}^{{\dot{h}}}{{,}^{{\dot{h}}}}\dot{R}_{{\dot{\upsilon }}}^{{\dot{A}}},\dot{\tau }_{{\dot{B}}}^{{\dot{h}}},\dot{H}_{f}^{{\dot{h}}}$,... need no further identification. In the dual context in which $\dot{h}$ is coparabolic we write $\mathop{G}\limits^{*} M_{{\dot{D}}}^{{\dot{h}}}{{,}^{{\dot{h}}}}{{\mathop{{{\text{ }}R}}\limits_{{\dot{\upsilon }}}^{*} }^{{\dot{A}}}},\mathop{\tau }\limits_{{{{{\dot{B}}}^{{\dot{h}}}}}}^{*} ,\mathop{{H_{f}^{{\dot{h}}}}}\limits^{*} , \ldots$