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Weak and strong solvability of parabolic variational inequalities in Banach spaces

Authors:

Matthew?Rudd rudd@mathutexasedu" title="rudd@mathutexasedn Email author" target="_blank">rudd@mathutexasedu" itemprop="email" data-track="click" data-track-action="Email author" data-track-label="">Email author

Institution:

(1) Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112-0090, USA;(2) Present address: Department of Mathematics, University of Texas at Austin, Austin, TX, 78712

Abstract:

We consider parabolic variational inequalities having the strong formulation

$\left\{ {\begin{array}{*{20}c} {\left\langle {u'(t),\,v - \left. {u(t)} \right\rangle + \left\langle {Au(t),} \right.\,v - \left. {u(t)} \right\rangle + \Phi (v) - \Phi (u(t) \geq 0,} \right.} \\ {\forall v \in V^{**} ,\,a.e.\,t \geq 0,} \\ \end{array} } \right.$

((1))

where

for some admissible initial datum, V is a separable Banach space with separable dual $V^* ,A:V^{**} \to V^*$ is an appropriate monotone operator, and $\Phi :V^{**} \to \mathbb{R} \cup \{ \infty \}$ is a convex, ${\text{weak}}^*$ lower semicontinuous functional. Well-posedness of (1) follows from an explicit construction of the related semigroup $\{ S(t):t \geq 0\} .$ Illustrative applications to free boundary problems and to parabolic problems in Orlicz-Sobolev spaces are given.

Keywords:

Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 35 47D 49

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