Degenerate Free Discontinuity Problems and Spectral Inequalities in Quantitative Form

Second Order Variational Problems with Free Discontinuity and Free Gradient Discontinuity

The Calibration Method for Free Discontinuity Problems

The calibration method is used to identify some minimizers of the Mumford-Shah functional. The method is then extended to more general free discontinuity problems.

Regular Minimizers of Some Free Discontinuity Problems

We show that minimizers of free discontinuity problems with energy dependent on jump integrals and Dirichlet boundary conditions are smooth provided a smallness condition is imposed on data. We examine several examples, including elastic-plastic beam and plate with free yield lines and deformable body with free damage. In all cases there is a gap between the condition for solvability (safe load...

Finite Difference Approximation of Free Discontinuity Problems

We approximate functionals depending on the gradient of u and on the behaviour of u near the discontinuity points, by families of non-local functionals where the gradient is replaced by finite differences. We prove pointwise convergence, Γ-convergence, and a compactness result which implies, in particular, the convergence of minima and minimizers. AMS (MOS) subject classifications: 49J45, 65K10.

Γ-convergence of Free-discontinuity Problems

We study the Γ-convergence of sequences of free-discontinuity functionals depending on vector-valued functions u which can be discontinuous across hypersurfaces whose shape and location are not known a priori. The main novelty of our result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Further, we co...