Epsilon-regularity for almost-minimizers of anisotropic free interface problem with Hölder dependence on the position

We establish an ε-regularity result for almost-minimizers of a class of variational problems involving both bulk and interface energies. The bulk energy is of Dirichlet type. The surface energy exhibits anisotropic behavior and is defined by means of an ellipsoidal density that is Hölder continuous with respect to the position variable.