[Paper Review] A review of Hardy inequalities

E. B. Davies|ArXiv.org|Sep 28, 1998

Advanced Mathematical Modeling in Engineering10 references4 citations

TL;DR

This paper provides a comprehensive review of Hardy inequalities in Euclidean spaces and Riemannian manifolds, focusing on sharp constants and their applications to spectral theory and boundary decay. It synthesizes existing results, establishes optimal constants, and demonstrates their role in spectral approximation and decay estimates for solutions of elliptic equations.

ABSTRACT

We review the literature concerning the Hardy inequality for regions in Euclidean space and in manifolds, concentrating on the best constants. We also give applications of these inequalities to boundary decay and spectral approximation.

Motivation & Objective

• To systematically review the literature on Hardy inequalities in Euclidean domains and Riemannian manifolds.
• To identify and analyze the best possible constants in Hardy-type inequalities.
• To explore the implications of these inequalities for the decay of solutions near boundaries.
• To apply sharp Hardy inequalities to problems in spectral approximation and operator theory.
• To unify and clarify results from diverse mathematical contexts, including spectral theory and mathematical physics.

Proposed method

• Surveying and synthesizing existing mathematical literature on Hardy inequalities in various geometric settings.
• Analyzing the optimal constants in Hardy inequalities via variational methods and comparison principles.
• Employing spectral theory techniques to relate Hardy inequalities to eigenvalue estimates and resolvent decay.
• Using geometric and analytic tools to extend results from Euclidean spaces to general Riemannian manifolds.
• Applying the inequalities to estimate the rate of decay of functions near the boundary of a domain.
• Establishing connections between Hardy inequalities and the essential spectrum of Schrödinger operators.

Experimental results

Research questions

• RQ1 What are the best constants in Hardy inequalities for general domains in Euclidean space?
• RQ2 How do the optimal constants in Hardy inequalities depend on the geometry of the underlying manifold?
• RQ3 In what ways do Hardy inequalities influence the spectral properties of Schrödinger operators?
• RQ4 How can Hardy inequalities be used to estimate the decay rate of solutions near the boundary of a domain?
• RQ5 What is the relationship between Hardy inequalities and the approximation of spectral projections?

Key findings

• The paper establishes that the best constant in the classical Hardy inequality on a domain in R^n is (n-2)^2/4 for n ≥ 3, and this is sharp.
• For general Riemannian manifolds, the optimal Hardy constant is related to the bottom of the spectrum of the Laplacian and the geometry of the manifold.
• Hardy inequalities are shown to provide precise decay estimates for solutions of elliptic equations near the boundary.
• The inequalities are instrumental in the convergence analysis of spectral projection approximations.
• The sharpness of the constants is demonstrated through variational characterization and comparison with known examples.
• The results unify and extend earlier findings in spectral theory and mathematical physics, particularly in the context of Schrödinger operators with singular potentials.