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Transactions of the American Mathematical Society

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

 

 

Regularity of weak solutions of parabolic variational inequalities


Author: William P. Ziemer
Journal: Trans. Amer. Math. Soc. 309 (1988), 763-786
MSC: Primary 35D10; Secondary 35K85, 49A29
DOI: https://doi.org/10.1090/S0002-9947-1988-0961612-9
MathSciNet review: 961612
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Abstract: In this paper, parabolic operators of the form

$\displaystyle {u_t} - \operatorname{div} A(x,\,t,\,u,\,Du) - B(x,\,t,\,u,\,Du)$

are considered where $ A$ and $ B$ are Borel measurable and subject to linear growth conditions. Let $ \psi :\,\Omega \to {R^1}$ be a Borel function bounded above (an obstacle) where $ \Omega \subset {R^{n + 1}}$. Let $ u \in {W^{1,2}}(\Omega )$ be a weak solution of the variational inequality in the following sense: assume that $ u \geqslant \psi $ q.e. and

$\displaystyle \int_\Omega {{u_t}\varphi + A \cdot D\varphi - B\varphi \geqslant 0} $

whenever $ \varphi \in W_0^{1,2}(\Omega )$ and $ \varphi \geqslant u - \psi $ q.e. Here q.e. means everywhere except for a set of classical parabolic capacity. It is shown that $ u$ is continuous even though the obstacle may be discontinuous. A mild condition on $ \psi $ which can be expressed in terms of the fine topology is sufficient to ensure the continuity of $ u$. A modulus of continuity is obtained for $ u$ in terms of the data given for $ \psi $.

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DOI: https://doi.org/10.1090/S0002-9947-1988-0961612-9
Article copyright: © Copyright 1988 American Mathematical Society