Sharp spectral estimates for free boundary problems arising in plasma physics

TL;DR

This paper establishes a precise spectral estimate for a free boundary problem in plasma physics, analyzing a non-local eigenvalue and its positivity across dimensions, with implications for the Emden equation.

Contribution

It provides a novel sharp spectral estimate for a superlinear free boundary problem, including analysis of a non-local eigenvalue and its positivity in multiple dimensions.

Findings

The first eigenvalue $\sigma_1$ is always positive in any dimension $N \\geq 2$.

The eigenvalue $\sigma_1$ does not satisfy a Faber-Krahn type isoperimetric property.

Implications for the uniqueness of solutions to the Emden equation are discussed.

Abstract

We derive a sharp spectral estimate for a superlinear free boundary problem arising in plasma physics. The semilinear equation is coupled with a constraint, which forces the analysis of a non-local eigenvalue equation. Consequently the corresponding first eigenvalue, say $\sigma_1$, is not a standard one and it is shown that it cannot satisfy a general isoperimetric property of Faber-Krahn type. This motivates a careful analysis of the problem on balls in any dimension $N\geq 2$, where we prove that in fact $\sigma_1$ is always positive. The implications about the uniqueness problem for the Emden equation are also discussed.