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

Gtit 

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Copyright information

© Springer-Verlag New York Inc. 1984

Authors and Affiliations

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

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