  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 $n \geq 2$.

Analysis of PDEs