Nagoya Mathematical Journal

On a regularity property and a priori estimates for solutions of nonlinear parabolic variational inequalities

Haruo Nagase

Abstract

In this paper we consider the following nonlinear parabolic variational inequality; $u(t) \in D(\Phi)$ for all $t \in I$, $(u_t(t), u(t) - v) + \langle \Delta_p u(t), u(t) - v \rangle + \Phi(u(t)) - \Phi(v) \leqq (f(t), u(t) -v )$ for all $v \in D(\Phi)$ a.e. $t\in I$, $u(x,0) = u_0(x)$, where $\Delta_p$ is the so-called $p$-Laplace operator and $\Phi$ is a proper, lower semicontinuous functional. We have obtained two results concerning to solutions of this problem. Firstly, we prove a few regularity properties of solutions. Secondly, we show the continuous dependence of solutions on given data $u_0$ and $f$.

Article information

Source
Nagoya Math. J., Volume 160 (2000), 123-134.

Dates
First available in Project Euclid: 27 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1114631501

Mathematical Reviews number (MathSciNet)
MR1804140

Zentralblatt MATH identifier
1007.35044

Citation

Nagase, Haruo. On a regularity property and a priori estimates for solutions of nonlinear parabolic variational inequalities. Nagoya Math. J. 160 (2000), 123--134. https://projecteuclid.org/euclid.nmj/1114631501

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