Characterizing compact coincidence sets in the thin obstacle problem and the obstacle problem for the fractional Laplacian

TL;DR

This paper classifies global solutions of the fractional Laplacian obstacle problem with compact coincidence sets, linking solutions to polynomial asymptotics and establishing convexity for solutions with quadratic growth.

Contribution

It provides a complete classification of solutions with compact coincidence sets and relates their asymptotics to polynomials, advancing understanding of the fractional obstacle problem.

Findings

Classification of global solutions with polynomial growth

Bijection between solutions and polynomial asymptotics

Convexity of compact coincidence sets for quadratic growth solutions

Abstract

In this paper we give a full classification of global solutions of the obstacle problem for the fractional Laplacian (including the thin obstacle problem) with compact coincidence set and at most polynomial growth in dimension $N \geq 3$. We do this in terms of a bijection onto a set of polynomials describing the asymptotics of the solution. Furthermore we prove that coincidence sets of global solutions that are compact are also convex if the solution has at most quadratic growth.