The variable coefficient thin obstacle problem: Carleman inequalities

Abstract

In this article we present a new strategy of addressing the (variable coefficient) thin obstacle problem. Our approach is based on a (variable coefficient) Carleman estimate. This yields semi-continuity of the vanishing order, lower and uniform upper growth bounds of solutions and sufficient compactness properties in order to carry out a blow-up procedure. Moreover, the Carleman estimate implies the existence of homogeneous blow-up limits along certain sequences and ultimately leads to an almost optimal regularity statement. As it is a very robust tool, it allows us to consider the problem in the setting of Sobolev metrics, i.e. working on the upper half ball B1+⊂R+n+1, the coefficients are only W1,p regular for some p>n+1.These results provide the basis for our further analysis of the free boundary, the optimal regularity of solutions and a first order asymptotic expansion of solutions at the regular free boundary which is carried out in a follow-up article, [21], in the framework of W1,p, p>2(n+1), regular coefficients and W2,p, p>2(n+1), regular non-zero obstacles.