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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Exceptional boundary sets for solutions of parabolic partial differential inequalities

Authors: G. N. Hile and R. Z. Yeh
Journal: Trans. Amer. Math. Soc. 306 (1988), 607-621
MSC: Primary 35K10; Secondary 35B05
MathSciNet review: 933309
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Abstract: Let $ \mathcal{M}$ be a second order, linear, parabolic partial differential operator with coefficients defined in a domain $ \mathcal{D} = \Omega \times (0,\,T)$ in $ {{\mathbf{R}}^n} \times {\mathbf{R}}$, with $ \Omega $ a domain in $ {{\mathbf{R}}^n}$. Let $ u$ be a suitably regular real function in $ \mathcal{D}$ such that $ u$ is bounded below and $ \mathcal{M}u$ is bounded above in $ \mathcal{D}$. If $ u \geqslant 0$ on $ \Omega \times \{ 0\} $ except on a set $ \Gamma \times \{ 0\} $, with $ \Gamma $ a subset of $ \Omega $ of suitably restricted Hausdorff dimension, then necessarily $ u \geqslant 0$ also on $ \Gamma \times \{ 0\} $. The allowable Hausdorff dimension of $ \Gamma $ depends on the coefficients of $ \mathcal{M}$. For example, if $ \mathcal{M}$ is the heat operator $ \Delta - \partial /\partial t$, the Hausdorff dimension of $ \Gamma $ needs to be smaller than the number of space dimensions $ n$.

Analogous results are valid for exceptional boundary sets on the lateral boundary, $ \partial \Omega \times (0,\,T)$, of $ \mathcal{D}$.

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Keywords: Parabolic operator, Phragmén-Lindelöf principle, exceptional boundary set, Hausdorff dimension
Article copyright: © Copyright 1988 American Mathematical Society