Improved Rellich inequalities for the polyharmonic operator

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.

Theorems & Definitions (10)

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