[Paper Review] Hardy inequalities with double singular weights

Nikolai Kutev, Tsviatko Rangelov|arXiv (Cornell University)|Jan 21, 2020

Advanced Mathematical Modeling in Engineering65 references4 citations

TL;DR

This paper establishes new sharp Hardy inequalities with double singular weights—singularities at both an interior point and the boundary of a domain—enabling improved analytical lower bounds for the first eigenvalue of the p-Laplacian. The method leverages optimal constants and nonlinear boundary terms to achieve sharpness, outperforming existing estimates in accuracy and generality across dimensions and p-values beyond 1.

ABSTRACT

The aim of this paper is to obtain new Hardy inequalities with double singular weights - at an interior point and on the boundary of the domain. These inequalities give us the possibility to derive estimates from below of the first eigenvalue of the p-Laplacian with Dirichlet boundary conditions.

Motivation & Objective

• To derive Hardy inequalities with optimal constants for domains featuring singular weights at both an interior point and the boundary.
• To improve analytical lower estimates of the first eigenvalue of the p-Laplacian in bounded domains, particularly for p>1 and n≥2.
• To establish sharpness of the inequalities through nonlinear additional terms, where equality is achieved for specific functions.
• To demonstrate that double singular weights yield tighter lower bounds than single-weight or alternative methods.
• To provide a general framework applicable to star-shaped and general domains, extending existing results in the literature.

Proposed method

• Derives abstract Hardy inequalities with general weights, introducing a one-parametric family of inequalities to optimize constants.
• Applies variational techniques and comparison principles to prove optimality of the Hardy constant γ^p in star-shaped and general domains.
• Introduces a nonlinear additional boundary term to achieve sharpness, where equality is attained for certain admissible functions.
• Uses the p-Laplacian eigenvalue problem as a key application, linking the inequality to lower bounds on λ_{p,n}(Ω).
• Employs a comparison framework with existing methods, including Cheeger’s constant, Picone’s identity, and Sobolev inequalities.
• Validates results numerically by comparing analytical lower bounds with iterative numerical approximations from Biezuner et al. (2008).

Experimental results

Research questions

• RQ1 Can Hardy inequalities with singular weights at both an interior point and the boundary yield improved lower bounds for the first eigenvalue of the p-Laplacian?
• RQ2 What is the optimal constant in Hardy inequalities when singularities are present at both an interior point and the boundary of a domain?
• RQ3 How can sharpness of the inequality be achieved when the constant is optimal, given that equality is not achieved with linear terms?
• RQ4 To what extent do double singular weights outperform single-weight or other classical methods in estimating λ_{p,n}(Ω)?
• RQ5 What is the quantitative improvement of the new analytical lower bounds over existing estimates, particularly for p>1 and n≥2?

Key findings

• The proposed Hardy inequality with double singular weights yields analytical lower bounds for λ_{p,n}(B_R) that are consistently tighter than those from single-weight inequalities or other classical methods.
• For the unit ball B_1, the new estimate Λ^{(3,0)}_{p,n}(B_1) is approximately 2.5 times smaller than the numerical value Λ^{(num)}_{p,n}(B_1) across all tested p>1 and n≥2.
• The Hardy constant γ^p is proven optimal, meaning the inequality fails for any larger constant, and sharpness is achieved via a nonlinear boundary term.
• In star-shaped domains, the method achieves optimal constants and sharp inequalities, with equality attained for specific functions due to the nonlinear term.
• The framework generalizes to both star-shaped and general bounded domains, providing a unified approach to eigenvalue estimation.
• Numerical comparisons confirm that the new method provides analytical estimates valid for all p>1 and n≥2, with consistent improvement over existing benchmarks.