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

Keywords

Boundary Point Parabolic Function Nonempty Open Subset Classical Context Parabolic Regularity
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag New York Inc. 1984

Authors and Affiliations

• J. L. Doob
• 1
1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA