Regularity results for a free interface problem with Hölder coefficients

We study a class of variational problems involving both bulk and interface energies. The bulk energy is of Dirichlet type albeit of very general form allowing the dependence from the unknown variable u and the position x . We employ the regularity theory of Λ -minimizers to study the regularity of the free interface. The hallmark of the paper is the mild regularity assumption concerning the dependence of the coefficients with respect to x and u that is of Hölder type.