Quasilinear parabolic variational inequalities with multivalued lowerorder terms
Abstract
In this paper, we provide an analytical frame work for the following multivalued parabolic variational inequality in a cylindrical domain : Find and an such that where is some closed and convex subset, A is a timedependent quasilinear elliptic operator, and the multivalued function is assumed to be upper semicontinuous only, so that Clarke's generalized gradient is included as a special case. Thus, parabolic variationalhemivariational inequalities are special cases of the problem considered here. The extension of parabolic variationalhemivariational inequalities to the general class of multivalued problems considered in this paper is not only of disciplinary interest, but is motivated by the need in applications. The main goals are as follows. First, we provide an existence theory for the abovestated problem under coercivity assumptions. Second, in the noncoercive case, we establish an appropriate subsupersolution method that allows us to get existence, comparison, and enclosure results. Third, the order structure of the solution set enclosed by subsupersolutions is revealed. In particular, it is shown that the solution set within the sector of subsupersolutions is a directed set. As an application, a multivalued parabolic obstacle problem is treated.
 Publication:

Zeitschrift Angewandte Mathematik und Physik
 Pub Date:
 October 2014
 DOI:
 10.1007/s0003301303576
 Bibcode:
 2014ZaMP...65..845C