Regularity of the free boundary in the biharmonic obstacle problem (1603.06819v2)

Abstract: In this article we use flatness improvement argument to study the regularity of the free boundary for the biharmonic obstacle problem with zero obstacle. Assuming that the solution is almost one-dimensional, and that the non-coincidence set is an non-tangentially accessible (NTA) domain, we derive the $C^{1,\alpha}$-regularity of the free boundary in a small ball centered at the origin. From the $C^{1,\alpha}$-regularity of the free boundary we conclude that the solution to the biharmonic obstacle problem is locally $ C^{3,\alpha}$ up to the free boundary, and therefore $C^{2,1} $. In the end we study an example, showing that in general $ C^{2,\frac{1}{2}}$ is the best regularity that a solution may achieve in dimension $n \geq 2$.