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.Apu 0 q :" tu P W 2,2 pΩq : u ´u0 P W 1,2 0 pΩqu