Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon
type: Item
instance of: scholarly article
title: Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon
zbMATH Open document ID: 1298.47057
author: Baptiste Devyver
author: Martin Fraas
author: Yehuda Pinchover
published in: Journal of Functional Analysis
publication date: 2014-07-25T00:00:00Z
full work available at URL: https://arxiv.org/abs/1208.2342
review text: Let \(P\) be a nonnegative second-order linear elliptic operator with real coefficients, which are defined on a domain or on a noncompact manifold in \(\mathbb{R}^n\). The authors study the problem of finding as large as possible (in an appropriate sense) weight functions \(W\geq 0\) for weighted Hardy-type inequalities involving the operator \(P\) and punctured domains of the type \(\Omega\setminus \{0\}\), with \(0\in \Omega\). When \(P\) is symmetric, the Hardy inequality with weight \(W\) on a domain \(\Omega\) has the following form \[ \displaystyle{q(\varphi)\geq \lambda \int_\Omega W(x)|\varphi(x)|^2\,dx} \text{ for all }\varphi\in C_0^\infty(\Omega),\tag{1} \] where \(\lambda\) is a positive constant and \(q\) is the quadratic form associated to \(P\). The operator \(P\) is said to be subcritical when inequality (1) holds with a nonzero weight \(W\), while \(P\) is said to be critical when inequality \((1)\) holds if and only if \(W=0\). For a subcritical operator \(P\) and a nonzero weight \(W\) satisfying \((1)\), the authors introduce the following further criticality condition: let \(\lambda_0>0\) be the best constant for inequality (1). The operator \(P-\lambda_0 W\) is said to be null-critical in \(\Omega\) if the corresponding Rayleigh-Ritz variational problem admits no minimizers. Then the authors extend the previous criticality conditions to a general (nonsymmetric) nonnegative operator \(P\) and give the definition of optimal Hardy-weight \(W\) in terms of the criticality properties of the operator \(P-\lambda W\), where \(\lambda\geq 1\). The main result of this paper states the existence of optimal Hardy-weights for nonnegative subcritical second-order linear elliptic operators. In particular, the authors prove that an optimal Hardy-weight for the operator \(Pu=-a_{ij}(x)\partial_i\partial_j u+b(x)\nabla u+c(x)u\) is \(W=\frac{1}{4}|\nabla \log(G/u)|_A^2\), where \(G\) is the minimal positive Green function of \(P\), \(u\) is a positive solution …
reviewed by: Giovanni Anello
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 35J15
Mathematics Subject Classification ID: 35A15
zbMATH DE Number: 6320687
zbMATH Keywords: Agmon metric
zbMATH Keywords: ground state
zbMATH Keywords: Hardy inequality
zbMATH Keywords: logarithmic Caccioppoli inequality
zbMATH Keywords: minimal growth
zbMATH Keywords: positive solutions
zbMATH Keywords: Rellich inequality
zbMATH Keywords: weighted Poincaré inequality
MaRDI profile type: Publication
OpenAlex ID: W2043985822
arXiv ID: 1208.2342
cites work: Best constants and Pohozaev identity for Hardy-Sobolev-type operators
cites work: Role of the fundamental solution in Hardy—Sobolev-type inequalities
cites work: Q3969238
cites work: Q3727040
cites work: Q3708420
cites work: Some results and examples about the behavior of harmonic functions and Green’s functions with respect to second order elliptic operators
cites work: Hardy's inequality and curvature
cites work: A unified approach to improved $L^p$ Hardy inequalities with best constants
cites work: Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators
cites work: Hardy inequalities on non-compact Riemannian manifolds
cites work: Improved Multipolar Hardy Inequalities
cites work: Optimal Hardy inequalities for general elliptic operators with improvements
cites work: Q5489533
cites work: On the finiteness of the Morse index for Schrödinger operators
cites work: A spectral result for Hardy inequalities
cites work: Optimal Hardy-type inequalities for elliptic operators
cites work: Optimal \(L^{p}\) Hardy-type inequalities
cites work: Critical points of Green's functions on complete manifolds
cites work: Hardy-Poincaré inequalities with boundary singularities
cites work: Optimizing improved Hardy inequalities
cites work: On the structure of Hardy-Sobolev-Maz'ya inequalities
cites work: On complete minimal surfaces with finite Morse index in three manifolds
cites work: Perturbative Oscillation Criteria and Hardy-Type Ineqalities
cites work: Bessel pairs and optimal Hardy and Hardy-Rellich inequalities
cites work: Functional Inequalities: New Perspectives and New Applications
cites work: Hardy and Rellich-type inequalities for metrics defined by vector fields
cites work: Q5294569
cites work: On a lower bound for the first eigenvalue of the Laplace operator on a riemannian manifold
cites work: Q4735474
cites work: Improved Hardy and Rellich inequalities on Riemannian manifolds
cites work: Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l'hydrodynamique
cites work: A geometric characterization of a sharp Hardy inequality
cites work: Weighted Poincaré inequality and rigidity of complete manifolds
cites work: Positive solutions to singular semilinear elliptic equations with critical potential on cone-like domains
cites work: Q5519383
cites work: On the best constant for Hardy’s inequality in $\mathbb {R}^n$
cites work: Structure of positive solutions to \((-\Delta +V)u=0\) in \(R^ n\)
cites work: Semismall perturbations in the Martin theory for elliptic equations
cites work: On variational principles for the generalized principal eigenvalue of second order elliptic operators and some applications
cites work: On criticality and ground states of second order elliptic equations. II
cites work: Large Scale Properties of Multiparameter Oscillation Problems
cites work: On positive Liouville theorems and asymptotic behavior of solutions of Fuchsian type elliptic operators
cites work: On positivity, criticality, and the spectral radius of the shuttle operator for elliptic operators
cites work: A Liouville-type theorem for Schrödinger operators
cites work: Q5427962
cites work: Existence of minimizers for Schroedinger operators under domain perturbations with application to Hard's inequality
cites work: A ground state alternative for singular Schrödinger operators
cites work: Ground state alternative for \(p\)-Laplacian with potential term
cites work: Q4318497
DOI: 10.1016/J.JFA.2014.01.017
description: scientific article