Local T-sets and renormalized solutions of degenerate quasilinear elliptic equations with an \(L^ 1\)-datum
The questions of uniqueness and stability of renormalized solutions for a boundary value problem associated to the equation \[ Au= -\text{div}(\widehat a(x,u,\nabla u))+ b(x)|u|^{\gamma-1}u=\mu\quad\text{on }\Omega \] with given \(\mu\in L^1(\Omega)\) are studied. Here \(\widehat a\) is a Carathéodory function such that \(\widehat a/a\) satisfies the general conditions of Leray-Lions, and \(\gamma\) is a real number. Because of the degeneracy of \(a\), the authors introduce the notion of local \(T\)-sets associated with the singularity \(S\) of \(\widehat a\). Following the same ideas as for usual Sobolev spaces, some useful properties of these sets and precise formulas and inequalities are given. The notion of renormalized solution is introduced. The uniqueness and stability for \(S\) satisfying a geometrical condition, or \(S\) satisfying an analytic-geometrical one, are given.
scientific article; zbMATH DE number 936929