The biharmonic Alt–Caffarelli problem in 2D

Abstract

We examine a variational free boundary problem of Alt–Caffarelli type for the biharmonic operator with Navier boundary conditions in two dimensions. We show interior C^2-regularity of minimizers and that the free boundary consists of finitely many C^2-hypersurfaces. With the aid of these results, we can prove that minimizers are in general not unique. We investigate radial symmetry of minimizers and compute radial solutions explicitly.