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Sharp estimates, uniqueness and spikes condensation for superlinear free boundary problems arising in plasma physics

Daniele Bartolucci, Aleks Jevnikar, Ruijun Wu

TL;DR

The paper analyzes superlinear free boundary problems of Grad–Shafranov type, recasting them into a parameterized system ${\textbf{(P}}_{\boldsymbol{\lambda}}{)}$ and studying the sharp energy structure, positivity thresholds, and uniqueness. In two dimensions, it delivers a sharp nonlinear generalization of Temam’s linear result, yielding a complete uniqueness result for all $p<\infty$ and clarifying the equality cases via minimizers of Sobolev-type functionals. In $N\ge3$ dimensions, the authors develop a Brezis–Merle type concentration-compactness framework and prove a spike condensation/quantization phenomenon, linking interior spikes to Green’s functions and revealing an infinite-mass regime that drives singular behavior. They also establish a dual variational formulation, provide a priori energy bounds, and, for balls, obtain a full radial uniqueness description with explicit parameterizations. Together, these results give a detailed global and local picture of spike formation, plasma regions, and the free boundary in Tokamak-relevant equilibria, with implications for both theory and plasma physics applications.

Abstract

We are concerned with Grad-Shafranov type equations, describing in dimension $N=2$ the equilibrium configurations of a plasma in a Tokamak. We obtain a sharp superlinear generalization of the result of Temam (1977) about the linear case, implying the first general uniqueness result ever for superlinear free boundary problems arising in plasma physics. Previous general uniqueness results of Beresticky-Brezis (1980) were concerned with globally Lipschitz nonlinearities. In dimension $N\geq 3$ the uniqueness result is new but not sharp, motivating the local analysis of a spikes condensation-quantization phenomenon for superlinear and subcritical singularly perturbed Grad-Shafranov type free boundary problems, implying among other things a converse of the results about spikes condensation in Flucher-Wei (1998) and Wei (2001). Interestingly enough, in terms of the "physical" global variables, we come up with a concentration-quantization-compactness result sharing the typical features of critical problems (Yamabe $N\geq 3$, Liouville $N=2$) but in a subcritical setting, the singular behavior being induced by a sort of infinite mass limit, in the same spirit of Brezis-Merle (1991).

Sharp estimates, uniqueness and spikes condensation for superlinear free boundary problems arising in plasma physics

TL;DR

The paper analyzes superlinear free boundary problems of Grad–Shafranov type, recasting them into a parameterized system and studying the sharp energy structure, positivity thresholds, and uniqueness. In two dimensions, it delivers a sharp nonlinear generalization of Temam’s linear result, yielding a complete uniqueness result for all and clarifying the equality cases via minimizers of Sobolev-type functionals. In dimensions, the authors develop a Brezis–Merle type concentration-compactness framework and prove a spike condensation/quantization phenomenon, linking interior spikes to Green’s functions and revealing an infinite-mass regime that drives singular behavior. They also establish a dual variational formulation, provide a priori energy bounds, and, for balls, obtain a full radial uniqueness description with explicit parameterizations. Together, these results give a detailed global and local picture of spike formation, plasma regions, and the free boundary in Tokamak-relevant equilibria, with implications for both theory and plasma physics applications.

Abstract

We are concerned with Grad-Shafranov type equations, describing in dimension the equilibrium configurations of a plasma in a Tokamak. We obtain a sharp superlinear generalization of the result of Temam (1977) about the linear case, implying the first general uniqueness result ever for superlinear free boundary problems arising in plasma physics. Previous general uniqueness results of Beresticky-Brezis (1980) were concerned with globally Lipschitz nonlinearities. In dimension the uniqueness result is new but not sharp, motivating the local analysis of a spikes condensation-quantization phenomenon for superlinear and subcritical singularly perturbed Grad-Shafranov type free boundary problems, implying among other things a converse of the results about spikes condensation in Flucher-Wei (1998) and Wei (2001). Interestingly enough, in terms of the "physical" global variables, we come up with a concentration-quantization-compactness result sharing the typical features of critical problems (Yamabe , Liouville ) but in a subcritical setting, the singular behavior being induced by a sort of infinite mass limit, in the same spirit of Brezis-Merle (1991).
Paper Structure (9 sections, 29 theorems, 347 equations)

This paper contains 9 sections, 29 theorems, 347 equations.

Table of Contents

  1. Introduction
  2. The proof of Theorem \ref{['thm1.1']}
  3. An estimate about best Sobolev constants
  4. Energy estimates for N>=3.
  5. Uniqueness for N>=3.
  6. A Brezis-Merle type result for subcritical problems in dimension N>2
  7. Asymptotics of free boundary problems
  8. Appendix A
  9. Appendix B

Key Result

Theorem 1.1

Let $N=2$ and $p\in [1,+\infty)$, then and the equality holds if and only if either $p=1$ or $p>1$, $\Omega=\mathbb{D}_2$ and $\lambda\psi_{ \lambda}$ is a minimizer of $\Lambda(\mathbb{D}_2,p+1)$.

Theorems & Definitions (50)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 1.3
  • Lemma 1.4
  • Remark 1.5
  • Theorem 1.6
  • Definition 1.7
  • Definition 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 40 more