John Andersson: Optimal Regularity for the Obstacle Problem

Abstract: In this talk we will consider the regularity for the obstacle problem without sign restriction. That is, we assume that $u \in W^{2,2}(B_1(0))$ and that $\Delta u(x) =f(x) \chi_{\{u\ne 0\}}$ where
$\chi_A$ is the indicator function for the set $A$ and $f(x)$ is a given, say Lipschitz, function.

Since $\Delta u$ is bounded it follows, from classical theory, that $u\in C^{1,\alpha}$ for every $\alpha<1$. But for the obstacle problem it has been known since the 60ies that the solution is actually in $C^{1,1}$ if $f$ is smooth and $u\ge 0$. In 2000 L.A. Caffarelli, L. Karp and H. Shahgholian (Ann. Math. 2000) showed that $u\in C^{1,1}$ – even when $u$ changes sign. This is an important condition needed to investigate the regularity of the set $\{n \ne 0\}$. Their proof was easily extendable to $f$ being Lipschitz.

This is an important result but it is unpleasant in certain ways that will be explained during the seminar. We will also sketch an elementary proof for a stronger result: that $u\in C^{1,1}$ even if $f$ is (just slightly better than) continuous as well as explain why this result is optimal. In describing the proof we will to touch upon some issues in singular integral operators at a basic level.

At the end of the seminar, if time permits, we will also discuss some possible extensions.

This talk is a joint work with H. Shahgholian and E. Lindgren (Comm. Pure Appl. Math 2013)