Communications in Analysis and Geometry

Volume 27 (2019)

Number 8

A fractional free boundary problem related to a plasma problem

Pages: 1665 – 1696



Mark Allen (Department of Mathematics, Brigham Young University, Provo, Utah, U.S.A.)


We study the solutions and the free boundary from a model that arises as a limit in a random homogenization of a fractional obstacle problem. This model also arises as a fractional analogue of a plasma problem from physics. Specifically, for a fixed bounded domain $\Omega$ we study solutions to the eigenfunction equation $(-\Delta)^s u = \lambda (u - \gamma)_{+}$ with $u \equiv 0$ on $\partial \Omega$. Our main object of study is the free boundary $\lbrace u = \gamma \rbrace$.

M. Allen was supported by NSF grant DMS-1303632.

Received 22 March 2016

Accepted 4 October 2017

Published 21 January 2020