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

In the present article we study a minimization problem in R-N involving the perimeter of the positivity set of the solution u and the integral of vertical bar del u vertical bar(p(x)). Here p(x) is a Lipschitz continuous function such that 1 < p(min <=) p(x) <= p(max) < infinity. 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 C-2 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 is an element of C-infinity we have that partial derivative(red) {u > 0} is a C-infinity surface.