Toward the $L^1$-theory of degenerate anisotropic elliptic variational inequalities

A. A. Kovalevsky

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg

Abstract: We consider nonlinear elliptic second-order variational inequalities with degenerate (with respect to the spatial variable) and anisotropic coefficients and $L^1$-data. We study the cases where the set of constraints belongs to a certain anisotropic weighted Sobolev space and a larger function class. In the first case, some new properties of $T$-solutions and shift $T$-solutions of the investigated variational inequalities are established. Moreover, the notion of $W^{1,1}$-regular $T$-solution is introduced, and a theorem of existence and uniqueness of such a solution is proved. In the second case, we introduce the notion of $\mathcal T$-solution of the variational inequalities under consideration and establish conditions of existence and uniqueness of such a solution.

Keywords: nonlinear elliptic variational inequalities; anisotropy; degeneration; $L^1$-data; $T$-solution; $\mathcal T$-solution.

UDC: 517.9

Received: 18.12.2014

 English version: Proceedings of the Steklov Institute of Mathematics (Supplement Issues), 2016, 292, suppl. 1, S156–S172

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