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Hardy inequalities for magnetic $p$-Laplacians

Cristian Cazacu, David Krejcirik, Nguyen Lam, Ari Laptev

TL;DR

The paper establishes improved Hardy-type inequalities for the magnetic $p$-Laplacian, showing that nontrivial magnetic fields render the operator subcritical in several regimes. It develops a gauge-invariant framework and leverages the diamagnetic inequality, exterior/interior Hardy bounds, and algebraic convexity properties to derive explicit weighted inequalities with unbounded weights, including a form with $\rho(x)=\frac{1}{|x|^d(|\log|x||^p+|x|^{p-d})}$ and another with $\rho(x)=\frac{1}{|x|^p(1+|\log|x||^p)}$. In two dimensions, Aharonov-Bohm fields yield a genuine improvement of the Hardy constant for $1<p<2$, as shown by a spectral analysis of the angular operator and a direct divergence-based proof. Collectively, these results extend the known $L^2$-magnetic Hardy improvements to the nonlinear $L^p$ setting and open several directions for optimal constants and further singular magnetic configurations.

Abstract

We establish improved Hardy inequalities for the magnetic $p$-Laplacian due to adding nontrivial magnetic fields. We also prove that for Aharonov-Bohm magnetic fields the sharp constant in the Hardy inequality becomes strictly larger than in the case of a magnetic-free $p$-Laplacian. We also post some remarks with open problems.

Hardy inequalities for magnetic $p$-Laplacians

TL;DR

The paper establishes improved Hardy-type inequalities for the magnetic -Laplacian, showing that nontrivial magnetic fields render the operator subcritical in several regimes. It develops a gauge-invariant framework and leverages the diamagnetic inequality, exterior/interior Hardy bounds, and algebraic convexity properties to derive explicit weighted inequalities with unbounded weights, including a form with and another with . In two dimensions, Aharonov-Bohm fields yield a genuine improvement of the Hardy constant for , as shown by a spectral analysis of the angular operator and a direct divergence-based proof. Collectively, these results extend the known -magnetic Hardy improvements to the nonlinear setting and open several directions for optimal constants and further singular magnetic configurations.

Abstract

We establish improved Hardy inequalities for the magnetic -Laplacian due to adding nontrivial magnetic fields. We also prove that for Aharonov-Bohm magnetic fields the sharp constant in the Hardy inequality becomes strictly larger than in the case of a magnetic-free -Laplacian. We also post some remarks with open problems.
Paper Structure (10 sections, 12 theorems, 109 equations)

This paper contains 10 sections, 12 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. The free $p$-Laplacian
  3. The magnetic $p$-Laplacian
  4. Preliminary lemmas
  5. Proof of Theorem \ref{['main_th_1']}
  6. Proof of Theorem \ref{['main_th_2']}
  7. Proof of Theorem \ref{['main_th_3']}
  8. Aharonov-Bohm magnetic fields
  9. Proof of Theorem \ref{['main_th_4']}
  10. Direct proof of Theorem \ref{['th3']}

Key Result

Proposition 1.1

Let $p\geq d$. If $V\in L_{\mathrm{loc}}^1(\mathbb{R}^d)$ is a non-negative potential such that then $V=0$ a.e. in $\mathbb{R}^d$.

Theorems & Definitions (25)

  • Definition 1.1
  • Proposition 1.1
  • Proposition 1.2
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1: Open problems
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.2: Open problem
  • Theorem 1.5: cf. Thm. 2.1.1, A
  • ...and 15 more