Hölder continuity conditions for the solvability of Dirichlet problems involving functionals with free discontinuities

The paper deals with the problem of minimizing a free discontinuity functional under Dirichlet boundary conditions. An existence result was known so far for C¹(∂Ω) boundary data û. We show here that the same result holds for ûC^0,μ(∂Ω) if and it cannot be extended to cover the case . The proof is based on some geometric measure theoretic properties, in part introduced here, which are proved a priori to hold for all the possible minimizers.