Local interpolation techniques for higher-order singular perturbations of non-convex functionals: free-discontinuity problems

We develop a general approach, using local interpolation inequalities, to non-convex integral functionals depending on the gradient with a singular perturbation by derivatives of order $k\ge 2$. When applied to functionals giving rise to free-discontinuity energies, such methods permit to change boundary values for derivatives up to order $k-1$ in problems defining density functions for the jump part, thus allowing to prove optimal-profile formulas, and to deduce compactness and lower bounds. As an application, we prove that for $k$-th order perturbations of energies depending on the gradient behaving as a constant at infinity, the jump energy density is a constant $m_k$ times the $k$-th root of the jump size. The result is first proved for truncated quadratic energy densities and in the one-dimensional case, from which the general higher-dimensional case can be obtained by slicing techniques. A wide class of non-convex energies can be studied as an envelope of these particular ones. Finally, we remark that an approximation of the Mumford-Shah functional can be obtained by letting $k$ tend to infinity. We also derive a new approximation of the Blake-Zisserman functional.