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Sharp spectral estimates for free boundary problems arising in plasma physics

Daniele Bartolucci, Aleks Jevnikar, Juncheng Wei, Ruijun Wu

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 $σ_1$, 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 $σ_1$ is always positive. The implications about the uniqueness problem for the Emden equation are also discussed.

Sharp spectral estimates for free boundary problems arising in plasma physics

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 , 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 , where we prove that in fact is always positive. The implications about the uniqueness problem for the Emden equation are also discussed.

Paper Structure

This paper contains 5 sections, 12 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Variational Solutions
  4. The proof of Theorem \ref{['thm:lm*']}: lambda*=infty
  5. On the uniqueness of solutions of the Emden problem

Key Result

Theorem 1.1

Let $p\in (1,p_N)$, then that is, for any $\lambda\geq 0$ and for any solution $(\alpha_{ \lambda},\psi_{ \lambda})$ of ${\textbf{(}\mathbf P\textbf{)}_{\mathbf \lambda}}$ on the ball $\mathbb{D}_N$ the inequality $\sigma_1(\alpha_{ \lambda},\psi_{ \lambda})>0$ holds true. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (21)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Proposition 2.7
  • proof
  • Theorem 3.1
  • ...and 11 more