Estimates for the gradient of solutions of elliptic equations in Orlicz-Sobolev spaces (Q2569692)

The author considers the Dirichlet problem \[ -\text{ div\,}(a(x,u,\nabla u))=f\quad \text{ in}\;\Omega;\qquad u=0\quad \text{ on}\;\partial\Omega \] in a smooth and bounded \(N\)-dimensional domain \(\Omega,\) \(N\geq2,\) where \[ a(x,s,\xi)\xi\geq A(|\xi|) \] with a convex function \(A(r),\) satisfying \(\lim_{r\to0^+} r^{-1}A(r)=0\) and \(\lim_{r\to+\infty} r^{-1}A(r)=+\infty.\) The problem with solely integrable right-hand side \(f\) is studied in the framework of suitable Sobolev-Orlicz spaces. Introducing the concept of entropy solutions, the author estimates their gradient and proves existence via symmetrization.