A Fractional Free Boundary Problem Related to a Plasma Problem

TL;DR

This paper investigates a fractional eigenvalue problem related to plasma physics, analyzing solutions to a nonlocal differential equation involving the fractional Laplacian and a positive part function within a bounded domain.

Contribution

It introduces a fractional analogue of a classical plasma problem, extending the understanding of free boundary problems to nonlocal operators.

Findings

Existence of solutions to the fractional plasma problem.

Characterization of free boundary behavior in the fractional setting.

Insights into spectral properties of the fractional operator.

Abstract

We study a fractional analogue of a plasma problem arising 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 $\partial \Omega$.