Free Boundary Problems and Density Perimeter

We introduce a new concept “the density perimeter,” which substitutes the measure of length of the perimeter of a set in free boundary problems. We deduce some links between the Hausdorff convergence and the char convergence for a family of domains with bounded density perimeter. If this perimeter is considered as a penalty term, we give existence results for the variational problem which appears in computer vision and in a Bernoulli-like free boundary problem. We also make some considerations concerning aΓ-convergence property of the density perimeter