Two phases problems with curvature: regularity of the free boundary Sandro SALSA Dept. of Mathematics, Politecnico di Milano

Abstract. We present some results on the regularity of the solution and of the free boundary in two phases problems where the mean curvature appears in the free boundary condition. Problems of this kind arise, for instance, in optimal design problems [1], in minimizing Dirichlet-Area functionals [2] or in Stefan-like problems where the speed of the free boundary depends on surface tension [3]. We discuss some of the ideas in [2] and in [3]. In particular, we prove that, for a rather general class of elliptic and evolution problems, Lipschitz free boundaries are smooth.

References

[1] L. Ambrosio, G. Buttazzo, An optimal design problem with perimeter penalization, Calc. Var. P.D.E. 1 (1993), no. 1, 55-69.

[2] I. Athanasopoulos, L. A. Caffarelli, C. Kenig, S. Salsa, An Area-Dirichlet Integral Minimization Problem, Comm. Pure Appl. Math. 54, 479-499 (2001).

[3] ———- Stefan like problems with curvature, to appear on Journal of Geometric Analysis.