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Improved Rellich inequalities for the polyharmonic operator

G. Barbatis

TL;DR

The paper sharpens Hardy–Rellich-type inequalities for the polyharmonic operator $(-\Delta)^m$ on convex domains by introducing two improvements: a logarithmic-series refinement and a geometric, volume-based refinement. The authors derive explicit constants $A(m)$, $B(m)$, and $\Gamma(m)$, prove the logarithmic-series bound is sharp with respect to $B(m)$, and extend the results to higher dimensions using a directional projection framework and Davies' mean-distance function. A separate, technically intricate analysis establishes the optimality of $B(m)$ via one-dimensional asymptotics and a careful limiting procedure. The outcomes unify and extend known cases ($m=1,2$) and provide tools for elliptic/parabolic PDEs with critical potentials on convex domains.

Abstract

We prove two improved versions of the Hardy-Rellich inequality for the polyharmonic operator $(-\Delta)^m$ involving the distance to the boundary. The first involves an infinite series improvement using logarithmic functions, while the second contains $L^2$ norms and involves as a coefficient the volume of the domain. We find explicit constants for these inequalities, and we prove their optimality in the first case.

Improved Rellich inequalities for the polyharmonic operator

TL;DR

The paper sharpens Hardy–Rellich-type inequalities for the polyharmonic operator on convex domains by introducing two improvements: a logarithmic-series refinement and a geometric, volume-based refinement. The authors derive explicit constants , , and , prove the logarithmic-series bound is sharp with respect to , and extend the results to higher dimensions using a directional projection framework and Davies' mean-distance function. A separate, technically intricate analysis establishes the optimality of via one-dimensional asymptotics and a careful limiting procedure. The outcomes unify and extend known cases () and provide tools for elliptic/parabolic PDEs with critical potentials on convex domains.

Abstract

We prove two improved versions of the Hardy-Rellich inequality for the polyharmonic operator involving the distance to the boundary. The first involves an infinite series improvement using logarithmic functions, while the second contains norms and involves as a coefficient the volume of the domain. We find explicit constants for these inequalities, and we prove their optimality in the first case.
Paper Structure (4 sections, 10 theorems, 99 equations)

This paper contains 4 sections, 10 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. One dimensional estimates
  3. Higher dimensions
  4. Optimality of the constants

Key Result

Theorem 1

Let \Omega be convex and such that d(x) is bounded in \Omega. Then there exists D\geq sup_{\Omega}d(x) such that for all functions u\in C^{\infty}_c(\Omega).

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 4
  • Lemma 5
  • Proposition 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Lemma 10