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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Exceptional boundary sets for solutions of parabolic partial differential inequalities
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by G. N. Hile and R. Z. Yeh PDF
Trans. Amer. Math. Soc. 306 (1988), 607-621 Request permission

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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Additional Information
  • © Copyright 1988 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 306 (1988), 607-621
  • MSC: Primary 35K10; Secondary 35B05
  • DOI: https://doi.org/10.1090/S0002-9947-1988-0933309-2
  • MathSciNet review: 933309