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The Rabinowitz continuum of subcritical Gelfand problems and free boundary-type equations arising in plasma physics

Daniele Bartolucci, Aleks Jevnikar, Juncheng Wei, Ruijun Wu

TL;DR

This work develops a novel energy-based, constrained formulation of subcritical Gelfand problems to elucidate the Rabinowitz continuum beyond minimal solutions. By transforming the problem into a Grad-Shafranov-type free boundary framework with density $\rho_{\lambda}=[\alpha_{\lambda}+\lambda\psi_{\lambda}]_+^p$ and energy $E_{\lambda}$, the authors construct a global, monotone parametrization of the continuum via the constrained branch $\mathcal{G}_0(\Omega)$ and its extensions, including a weighted spectral analysis that yields an unconditional uniqueness result for Grad-Shafranov-type equations on balls. They obtain sharp, radial descriptions on the ball, including explicit formulas in 2D and higher-dimensional analogues, and prove that the constrained branch can be continued up to the Rabinowitz turning point, producing a bell-shaped energy profile and a single turning point in the associated $\mu_{\lambda}$. The results provide a rigorous, general mechanism to describe the qualitative geometry of non-minimal Gelfand solutions on convex domains, with direct implications for plasma physics via Grad-Shafranov-type formulations. Overall, the paper delivers a robust 3- to 5-sentence synthesis of the continuum, a constructive parametrization, and precise monotonicity properties that extend classical minimal-solution theory to non-minimal branches.

Abstract

The qualitative behavior of the Rabinowitz unbounded continuum of subcritical Gelfand problems is well known on balls in any dimension. We don't know of any such sharp and detailed description otherwise, which is our motivation to look for a new approach to the problem. The underlying idea is to describe solutions of Gelfand problems via suitably defined constrained problems of free boundary-type arising in plasma physics and to replace the usual $L^\infty$ norm of the solution with the energy of the plasma. Toward this goal, we first solve a long standing open problem of independent interest about the uniqueness of solutions of Grad-Shafranov type equations. Thus, we exploit these unique solutions to detect a curve containing both minimal and non minimal solutions of the associated Gelfand problem. In other words we come up with a new global parametrization of the Rabinowitz continuum, the monotonicity of the energy along the branch providing a meaningful generalization of the classical pointwise monotonicity property of minimal solutions, suitable to describe non minimal solutions as well. On a ball in any dimension, we come up as expected with a bell-shaped profile of the full branch of solutions of the Gelfand problem.

The Rabinowitz continuum of subcritical Gelfand problems and free boundary-type equations arising in plasma physics

TL;DR

This work develops a novel energy-based, constrained formulation of subcritical Gelfand problems to elucidate the Rabinowitz continuum beyond minimal solutions. By transforming the problem into a Grad-Shafranov-type free boundary framework with density and energy , the authors construct a global, monotone parametrization of the continuum via the constrained branch and its extensions, including a weighted spectral analysis that yields an unconditional uniqueness result for Grad-Shafranov-type equations on balls. They obtain sharp, radial descriptions on the ball, including explicit formulas in 2D and higher-dimensional analogues, and prove that the constrained branch can be continued up to the Rabinowitz turning point, producing a bell-shaped energy profile and a single turning point in the associated . The results provide a rigorous, general mechanism to describe the qualitative geometry of non-minimal Gelfand solutions on convex domains, with direct implications for plasma physics via Grad-Shafranov-type formulations. Overall, the paper delivers a robust 3- to 5-sentence synthesis of the continuum, a constructive parametrization, and precise monotonicity properties that extend classical minimal-solution theory to non-minimal branches.

Abstract

The qualitative behavior of the Rabinowitz unbounded continuum of subcritical Gelfand problems is well known on balls in any dimension. We don't know of any such sharp and detailed description otherwise, which is our motivation to look for a new approach to the problem. The underlying idea is to describe solutions of Gelfand problems via suitably defined constrained problems of free boundary-type arising in plasma physics and to replace the usual norm of the solution with the energy of the plasma. Toward this goal, we first solve a long standing open problem of independent interest about the uniqueness of solutions of Grad-Shafranov type equations. Thus, we exploit these unique solutions to detect a curve containing both minimal and non minimal solutions of the associated Gelfand problem. In other words we come up with a new global parametrization of the Rabinowitz continuum, the monotonicity of the energy along the branch providing a meaningful generalization of the classical pointwise monotonicity property of minimal solutions, suitable to describe non minimal solutions as well. On a ball in any dimension, we come up as expected with a bell-shaped profile of the full branch of solutions of the Gelfand problem.

Paper Structure

This paper contains 10 sections, 21 theorems, 158 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary estimates for constrained solutions
  3. Spectral setup of the linearized problem
  4. Monotone parametrization of solutions
  5. The solution branch on the ball of unit volume
  6. Planar case N=2
  7. The higher dimensional case N>=3
  8. Differentiability of alpha-lambda
  9. The proof of Theorem \ref{['thm:lm*']}: $\lambda_*(\mathbb{D}_N,p)\geq \lambda_+(\mathbb{D}_N,p)$.
  10. Proof of Theorem \ref{['thm4']}

Key Result

Theorem 1.1

Let $N\geq 2$ and $p\in [1,p_N)$, then $\lambda_*(\Omega,p)\geq \lambda_0(\Omega,p)$ where the equality holds if and only if $p=1$ and for any $\lambda\in [0,\lambda_*(\Omega,p))$ there exists a unique solution to ${\textbf{(}\mathbf P\textbf{)}_{\mathbf \lambda}}$, defining a $C^1$ curve of solutio In particular $\mathcal{G}_0(\Omega)\subseteq \mathcal{G}_*(\Omega)$ where $\mathcal{G}_0(\Omega)\e

Figures (1)

  • Figure 1: The graph of $(\mu_{ \lambda},E_{ \lambda}),\,\lambda\in [0,\lambda_+(\mathbb{D}_{_N},p)]$. Here $\mu^t=\mu_\lambda\left.\right|_{\lambda=\lambda^t}$.

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof : Proof of Lemma \ref{['lemE1']}
  • Lemma 3.1
  • proof
  • Remark 3.2
  • ...and 30 more