A minimization problem for the p(x)-Laplacian involving area

Abstract

In the present article we study a minimization problem in RN involving the perimeter of the positivity set of the solution u and the integral of | ∇ u| p(x). Here p(x) is a Lipschitz continuous function such that 1 < pmin≤ p(x) ≤ pmax< ∞. We prove that such a minimizing function exists and that it is a classical solution to a free boundary problem. In particular, the reduced free boundary is a C2 surface and the dimension of the singular set is at most N- 8. Under further regularity assumptions on the exponent p(x) we get more regularity of the free boundary. In particular, if p∈ C∞ we have that ∂red{ u> 0 } is a C∞ surface.