This paper is devoted to study the existence of optimal domains which minimize the domain energy functional \(\Omega\to E(\Omega)\). Special cases are studied in which a family of open subsets of a fixed bounded set is compact in the char-topology defined on the family of measurable subsets of \(\mathbb{R}^n\) by the \(L^2\)-matrix, and in which the \(H^c\)-convergence implies the char-convergence at the same limit. The key technique is the introduction of the identity perimeter, as the authors call it, in free boundary problems in order to get zero measure boundaries. Putting a boundedness constraint on the density perimeter, the authors are able to prove that the \(H^c\)-convergence implies the char-convergence while each limiting term has zero boundary measure.
Finally, some existence results for Bernoulli like free boundary problems are proved by aid of a penalty method with density perimeter. For a variational problem arisen from the computer version, the authors prove the existence of the minimizing terms under the same penalty conditions and make some considerations involving a new concept: the maximal density curvature, precisely, they deduce the \(\Gamma\)-convergence of the density perimeter for \(\gamma\to 0\) to the Minkowski content in the family of closed sets with bounded maximal density curvature.