Quasilinear Parabolic Variational Inequalities with Multi-valued Lower-order Terms


Nipp, K.


In this paper, we provide an analytical frame work for the following multi-valued parabolic variational inequality in a cylindrical domain Q=Ω×(0,τ) : Find u∈K and an η∈Lp′(Q) such that η∈f(⋅,⋅,u),⟨ut+Au,v−u⟩+∫Qη(v−u)dxdt≥0,∀v∈K, where K⊂X0=Lp(0,τ;W1,p0(Ω)) is some closed and convex subset, A is a time-dependent quasilinear elliptic operator, and the multi-valued function s↦f(⋅,⋅,s) is assumed to be upper semicontinuous only, so that Clarke’s generalized gradient is included as a special case. Thus, parabolic variational–hemivariational inequalities are special cases of the problem considered here. The extension of parabolic variational–hemivariational inequalities to the general class of multi-valued 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 above-stated problem under coercivity assumptions. Second, in the noncoercive case, we establish an appropriate sub-supersolution method that allows us to get existence, comparison, and enclosure results. Third, the order structure of the solution set enclosed by sub-supersolutions is revealed. In particular, it is shown that the solution set within the sector of sub-supersolutions is a directed set. As an application, a multi-valued parabolic obstacle problem is treated.


Mathematics and Statistics

Keywords and Phrases

parabolic variational inequality; upper semicontinuous multi-valued operator; peudomonotone multi-valued operator; comparison principle; sub-ubersolution

Document Type

Article - Journal

Document Version


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© 2013 Springer Verlag, All rights reserved.

Publication Date

01 Jan 2013