The obstacle problem for a higher order fractional Laplacian

Abstract

In this paper, we consider the obstacle problem for the fractional Laplace operator $(-\Delta {)}^s$ in the Euclidian space ${\mathbb{R}}^n$ in the case where $1<s<2$. As first observed in \cite{Y}, the problem can be extended to the upper half-space ${\mathbb{R}}_{+}^{n+1}$ to obtain a thin obstacle problem for the weighted biLaplace operator ${\Delta }_b^{2}U$, where ${\Delta }_{b}U=y^{-b}\nabla \cdot (y^b\nabla U)$. Such a problem arises in connection with unilateral phenomena for elastic, homogenous, and isotropic flat plates. We establish the well-posedness and $C_{loc}^{1,1}({\mathbb{R}}^n)\cap H^{1+s}({\mathbb{R}}^n)$-regularity of the solution. By writing the solutions in terms of Riesz potentials of suitable local measures, we can base our proofs on tools from potential theory, such as a continuity principle and a maximum principle. Finally, we deduce the regularity of the extension problem to the higher dimensional upper half space. This gives an extension of Schild's work in \cite{Sc1} and \cite{Sc2} from the case $b=0$ to the general case $-1<b<1$.

26 pages