Weighted Hardy-Rellich inequalities via the Emden-Fowler transform

TL;DR

The work develops a sharp, parameter-dependent framework for weighted Hardy-Rellich inequalities on cone domains using the Emden–Fowler transform, translating power-weighted norms into weight-free cylindrical forms and reducing the problem to spectral data of the Laplace–Beltrami operator. The authors derive the optimal Hardy-Rellich constants $\mu_\alpha(C_\Sigma;\Lambda)$ as a minimum over $\lambda\in\Lambda$, establish conditions for positivity, and prove that extremals are not attained in the natural Sobolev spaces, with resonance phenomena depending on spectral placement. They further obtain improved inequalities with remainder terms under various boundary conditions (Navier–Dirichlet, Dirichlet) on cone-like domains, involving Leray-type and logarithmic weights, and provide sharp optimality statements for the remainder terms in several geometric configurations. The results encompass punctured space and half-space as particular cones, and extend to cone-like domains with precise angular spectral constraints, offering a comprehensive toolkit for sharp, geometry-dependent Rellich-type inequalities with optimal constants.

Abstract

We exploit a technique based on the Emden-Fowler transform to prove optimal Hardy-Rellich inequalities on cones, including the punctured space $\mathb{R}^N\setminus\{0\}$ and the half space as particular cases. We find optimal constants for classes of test functions vanishing on the boundary of the cone and possibly orthogonal to prescribed eigenspaces of the Laplace Beltrami operator restricted to the spherical projection of the cone. Furthermore, we show that extremals do not exist in the natural function spaces. Depending on the parameters, certain resonance phenomena can occur. For proper cones, this is excluded when considering test functions with compact support. Finally, for suitable subsets of the cones we provide improved Hardy-Rellich inequalities, under different boundary conditions, with optimal remainder terms.

Theorems & Definitions (34)

• Proposition 1.1
• Theorem 2.1
• Remark 2.2
• Lemma 2.3
• proof
• proof : Proof of Theorem \ref{['HR-cone']}
• Remark 2.4
• Remark 2.5
• Theorem 2.6
• proof