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 $$C^{1,\alpha }$$C1,α-regularity of the free boundary in a small ball centred at the origin. From the $$C^{1,\alpha }$$C1,α-regularity of the free boundary we conclude that the solution to the biharmonic obstacle problem is locally $$ C^{3,\alpha }$$C3,α up to the free boundary, and therefore $$C^{2,1}$$C2,1. In the end we study an example, showing that in general $$ C^{2,\frac{1}{2}}$$C2,12 is the best regularity that a solution may achieve in dimension $$n \ge 2$$n≥2.