Regularity of the free boundary in the biharmonic obstacle problem

In this article we use a 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 domain, we derive the C1,a-regularity of the free boundary in a small ball centred at the origin. From the C1,a-regularity of the free boundary we conclude that the solution to the biharmonic obstacle problem is locally C3,a up to the free boundary, and therefore C2,1. In the end we study an example, showing that in general C2, 1 2 is the best regularity that a solution may achieve in dimension n >= 2.