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Optimizers in Sobolev-curl inequalities

Jarosław Mederski, Andrzej Szulkin

TL;DR

This work analyzes a Sobolev-type inequality involving the $p$-curl operator in ${\mathbb{R}}^3$, establishing the existence of minimizers that yield ground-state solutions to the critical curl-curl equation ${\nabla}\times (|{\nabla}\times u|^{p-2}{\nabla}\times u)=|u|^{p^*-2}u$; it introduces a direct Nehari-type variational approach suited to strongly indefinite problems and proves strict positivity of the curl-curl Sobolev constant ${S}_{p,\mathrm{curl}}$ above a scaled Sobolev benchmark ${S}_p{\mathcal{H}}_p$. The paper develops a robust functional-analytic framework based on the Helmholtz decomposition, defines the manifolds ${\mathcal{M}}$ and ${\mathcal{N}}$, and proves attainment of the infimum on ${\mathcal{N}}$, yielding a ground-state solution and equality in the Sobolev-curl inequality. It further investigates symmetry by working in ${\mathcal{D}}^{1,p}_{\mathcal{O}}(\mathrm{curl};\mathbb{R}^3)$, obtaining symmetry-enhanced minimizers and, in the case $p=\tfrac{3}{2}$, two new nontrivial solutions, one of which is explicitly ${\mathcal{O}}$-equivariant. Finally, the authors present a novel route to compactness for Sobolev-type inequalities by projecting minimizing sequences onto a Nehari-type manifold and applying Solimini’s concentration result, providing an alternative proof of Lions’ concentration-compactness principle without invoking the full framework.

Abstract

We study a Sobolev-type inequality involving the $p$-curl operator in $\mathbb{R}^3$. We prove the existence of a minimizer which yields a solution to the $p$-curl-curl equation in the critical case. The problem is motivated both by nonlinear Maxwell equations and by the occurrence of zero modes in three-dimensional Dirac equations. Moreover, we introduce a new variational approach that allows to treat quasilinear strongly indefinite problems by direct minimization on a Nehari-type constraint. We also consider existence of minimizers under some symmetry assumptions. Finally, our approach offers a new proof of the compactness of minimizing sequences for the Sobolev inequalities in the critical case.

Optimizers in Sobolev-curl inequalities

TL;DR

This work analyzes a Sobolev-type inequality involving the -curl operator in , establishing the existence of minimizers that yield ground-state solutions to the critical curl-curl equation ; it introduces a direct Nehari-type variational approach suited to strongly indefinite problems and proves strict positivity of the curl-curl Sobolev constant above a scaled Sobolev benchmark . The paper develops a robust functional-analytic framework based on the Helmholtz decomposition, defines the manifolds and , and proves attainment of the infimum on , yielding a ground-state solution and equality in the Sobolev-curl inequality. It further investigates symmetry by working in , obtaining symmetry-enhanced minimizers and, in the case , two new nontrivial solutions, one of which is explicitly -equivariant. Finally, the authors present a novel route to compactness for Sobolev-type inequalities by projecting minimizing sequences onto a Nehari-type manifold and applying Solimini’s concentration result, providing an alternative proof of Lions’ concentration-compactness principle without invoking the full framework.

Abstract

We study a Sobolev-type inequality involving the -curl operator in . We prove the existence of a minimizer which yields a solution to the -curl-curl equation in the critical case. The problem is motivated both by nonlinear Maxwell equations and by the occurrence of zero modes in three-dimensional Dirac equations. Moreover, we introduce a new variational approach that allows to treat quasilinear strongly indefinite problems by direct minimization on a Nehari-type constraint. We also consider existence of minimizers under some symmetry assumptions. Finally, our approach offers a new proof of the compactness of minimizing sequences for the Sobolev inequalities in the critical case.

Paper Structure

This paper contains 8 sections, 15 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Functional setting and preliminaries
  3. Proof of Theorem \ref{['Th:main1']}
  4. Symmetry
  5. New approach to minimizing sequences in the Sobolev inequality
  6. Density lemma
  7. Concentraction-compactness result
  8. Relation between $v$ and $|v|$

Key Result

Theorem 1.1

(a) $S_{p,\mathrm{curl}} > S_p\cdot{\mathcal{H}}_p$. (b) If $(u_n)\subset {\mathcal{N}}$ is a minimizing sequence for $J$, then there are $(s_n)\subset (0,\infty)$ and $(y_n)\subset \mathbb{R}^3$ such that, passing to a subsequence, where $u$ is a minimizer for $J$ on ${\mathcal{N}}$. (c) $\inf_{{\mathcal{N}}}J=\frac{1}{3}S_{p,\mathrm{curl}}^{3/p}$ is attained, $u$ is a ground state solution to e

Theorems & Definitions (23)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • ...and 13 more