Nagoya Mathematical Journal

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

Haruo Nagase

Full-text: Open access

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

Subjects
Primary: 35K85: Linear parabolic unilateral problems and linear parabolic variational inequalities [See also 35R35, 49J40]
Secondary: 35B45: A priori estimates 35B65: Smoothness and regularity of solutions

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


Export citation

References

  • Y. Cheng, Hölder continuity of the inverse of $p$-Laplacian , J. Math. Anal. Appl., 221 (1998), 734–748.
  • J. Kacur, Nonlinear parabolic equations with the mixed nonlinear and nonstationary boundary conditions , Math. Slovaca, 30-3 (1980), 213–237.
  • ––––, On an approximate solution of variational inequalities , Math. Nachr., 123 (1985), 205–224.
  • ––––, Method of Rothe in Evolution Equations, Teubner-Texte zur Mathematik, 80, Leipzig (1985).
  • J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod Gauthier-Villars, Paris (1969).
  • H. Nagase, On an estimate for solutions of nonlinear elliptic variational inequalities , Nagoya Math. J., 107 (1987), 69–89.
  • ––––, On an application of Rothe's method to nonlinear parabolic variational inequalities , Funk. Ekv., 32-2 (1989), 273–299.
  • ––––, On an asymptotic behaviour of solutions of nonlinear parabolic variational inequalities , Japan. J. Math., 15-1 (1989), 169–189.
  • ––––, On some regularity properties for solutions of nonlinear parabolic differential equations , Nagoya Math. J., 128 (1992), 49–63.
  • ––––, A remark on decay properties of solutions of nonlinear parabolic variational inequalities , Japan. J. Math., 22 (1996), 285–292.
  • G. Savare, Weak solutions and maximal regularity for abstract evolution inequalities , Adv. Math. Sci. Appl., 6-2 (1996), 377–418.