Remote Access Transactions of the American Mathematical Society
Green Open Access

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
MathSciNet review: 961612
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35D10, 35K85, 49A29

Retrieve articles in all journals with MSC: 35D10, 35K85, 49A29

Additional Information

Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society