Regularity of the free boundary in the biharmonic obstacle problem

TL;DR

This paper investigates the regularity of the free boundary in the biharmonic obstacle problem, establishing conditions under which the boundary is $C^{1,eta}$ and solutions are $C^{2,1}$, with an example showing optimal regularity limits.

Contribution

It introduces a flatness improvement method to prove $C^{1,eta}$ regularity of the free boundary under specific assumptions, advancing understanding of solution smoothness in biharmonic obstacle problems.

Findings

Free boundary is $C^{1,eta}$ near regular points.

Solutions are locally $C^{2,1}$ up to the free boundary.

Optimal regularity in general is $C^{2,1/2}$ in higher dimensions.

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 $n \geq 2$.