This article is cited in 9 papers

On $T$-solutions of degenerate anisotropic elliptic variational inequalities with $L^1$-data

A. A. Kovalevsky^a, Yu. S. Gorban<sup>b

a</sup> Institute of Applied Mathematics and Mechanics, Ukraine National Academy of Sciences
^b Donetsk National University

Abstract: We introduce the notions of $T$-solutions and shift $T$-solutions of variational inequalities corresponding to a non-linear degenerate anisotropic elliptic operator, a constraint set in a sufficiently large class, and an $L^1$-right-hand side. We prove theorems on the existence and uniqueness of such solutions and describe their properties. While the notion of $T$-solution is defined only when the constraint set contains at least one bounded function, the notion of shift $T$-solution does not require this condition. We describe the relation between these notions and prove that these types of solutions of a variational inequality coincide with ordinary solutions whenever the right-hand side is sufficiently regular.

Keywords: degenerate anisotropic elliptic variational inequalities, $L^1$-data, $T$-solution, shift $T$-solution, existence and uniqueness of solutions.

UDC: 517.9

MSC: 35J85, 35J70, 35B65, 49J40

Received: 03.03.2008

DOI: 10.4213/im2774

 English version: Izvestiya: Mathematics, 2011, 75:1, 101–156

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