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

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.