# The Parabolic Dirichlet Problem, Sweeping, and Exceptional Sets

• J. L. Doob
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 262)

## 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 $$\mathop{G}\limits^{*} M_{{\dot{D}}}^{{\dot{h}}}{{,}^{{\dot{h}}}}{{\mathop{{{\text{ }}R}}\limits_{{\dot{\upsilon }}}^{*} }^{{\dot{A}}}},\mathop{\tau }\limits_{{{{{\dot{B}}}^{{\dot{h}}}}}}^{*} ,\mathop{{H_{f}^{{\dot{h}}}}}\limits^{*} , \ldots$$

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