Optimal Hardy-weights for the Finsler $p$-Dirichlet integral with a potential
Yongjun Hou
TL;DR
The paper develops a comprehensive framework for optimal Hardy-weights associated with the Finsler p-Dirichlet energy in anisotropic media, allowing a variable norm H(x,·) and a local Morrey-space potential V. Building on criticality theory and Bregman-distance techniques, it first establishes the zero-potential case, producing explicit optimal weights W_0 in terms of a positive Finsler p-harmonic function G and showing the ground state is tied to a simple power of G. It then extends to nonzero potentials by leveraging Green potentials G_φ to construct W that yield a critical, null-critical shifted functional Q_{V-W}, with the ground state given by f_0(G_φ). The results provide sharp, structurally explicit Hardy inequalities in both punctured domains and in the presence of potentials, with applications to eigenvalue problems and Morrey-space potentials. Overall, the work advances the theory of optimal Hardy-weights for nonlinear, anisotropic Dirichlet-type energies and broadens the toolkit for sharp functional-analytic inequalities in PDEs.
Abstract
Fix an integer $n\geq 2$, an exponent $1<p<\infty$, and a domain $Ω\subseteq\mathbb{R}^{n}$. Let $Ω^{*}\triangleqΩ\setminus\{\hat{x}\}$ where $\hat{x}\inΩ$. Under some further conditions, we construct optimal Hardy-weights for the Finsler $p$-Dirichlet integral $$Q_{0}[φ;Ω^{*}]\triangleq\int_{Ω^{*}}H(x,\nabla φ)^{p}\,\mathrm{d}x\quad \mbox{on}\quad C^{\infty}_{c}(Ω^{*}),$$ and the Finsler $p$-Dirichlet integral with a potential $$Q_{V}[φ;Ω]\triangleq\int_Ω\left(H(x,\nabla φ)^{p}+ V|φ|^{p}\right)\,\mathrm{d}x\quad \mbox{on}\quad C^{\infty}_{c}(Ω),$$where $H(x,\cdot)$ is a family of norms on $\mathbb{R}^{n}$ parameterized by $x\inΩ^{*}$ or $x\inΩ$, respectively, and the potential $V$ lies in a subspace $\widehat{M}^{q}_{\rm loc}(p;Ω)$ of a local Morrey space $M^{q}_{\rm loc}(p;Ω)$.