A construction of a Lipschitz continuous minimizer of a free boundary problem Original Research Article

The free boundary problem of the variational type studied by Alt and Caffarelli is investigated here. Especially, treated is the quadratic functional having the coefficient of diagonal form and the characteristic function term. The main result is to construct a minimizer with the Lipschitz continuity, which will play an essential role for proving the regularity of the free boundary. This is established by treating the regularized functional which approximates the original functional in the sense of Γ-convergence. The most important part of this paper is to show the uniform Lipschitz continuity of the minimizers of the regularized functionals. Taking advantage of the minimality, not resorting to the Euler–Lagrange equation, the minimizers turn out to belong to DeGiorgi class uniformly, the uniform Hölder estimate is obtained, and, as a result, the Lipschitz estimate is attained with the aid of the uniform Harnack inequality.