[Paper Review] Rellich inequalities with weights

Paolo Caldiroli, Roberta Musina|arXiv (Cornell University)|Mar 31, 2011

Nonlinear Partial Differential Equations2 references4 citations

TL;DR

This paper establishes the optimal constants in weighted Rellich inequalities on cone-like domains in $ ^n$, $n \geq 2$, for functions vanishing near the origin (Navier case) or with compact support (Dirichlet case). It derives explicit formulas for the best constants and proves improved inequalities with sharp logarithmic remainder terms, revealing a resonance phenomenon when $-\gamma_{n,\alpha}$ coincides with an eigenvalue of the Laplace-Beltrami operator on the sphere.

ABSTRACT

Let $Ω$ be a cone in $\mathbb{R}^{n}$ with $n\ge 2$. For every fixed $α\in\mathbb{R}$ we find the best constant in the Rellich inequality $\int_Ω|x|^α|Δu|^{2}dx\ge C\int_Ω|x|^{α-4}|u|^{2}dx$ for $u\in C^{2}_{c}(\barΩ\setminus\{0\})$. We also estimate the best constant for the same inequality on $C^{2}_{c}(Ω)$. Moreover we show improved Rellich inequalities with remainder terms involving logarithmic weights on cone-like domains.

Motivation & Objective

• To determine the best constant in weighted Rellich inequalities of the form $\int_\Omega |x|^\alpha |\Delta u|^2 \,dx \geq C \int_\Omega |x|^{\alpha-4} |u|^2 \,dx$ on cone-like domains.
• To analyze the Navier case ($u \in C^2_c(\overline{\Omega} \setminus \{0\})$) and the Dirichlet case ($u \in C^2_c(\Omega)$) separately, identifying the infimum of the Rayleigh quotient.
• To derive improved Rellich inequalities with remainder terms involving logarithmic weights on bounded and exterior cone-like domains, ensuring sharp constants.
• To characterize the failure of the inequality via spectral resonance: the inequality fails if $-\gamma_{n,\alpha}$ is an eigenvalue of $-\Delta_\sigma$ on $\mathbb{S}^{n-1}$.
• To provide a complete classification of the validity and sharpness of the inequality for all $\alpha \in \mathbb{R}$ and all dimensions $n \geq 2$.

Proposed method

• Use of dilation invariance to reduce the problem to homogeneous cones $\Omega = \mathcal{C}_\Sigma = \{ r\sigma \mid r > 0, \sigma \in \Sigma \}$, where $\Sigma \subset \mathbb{S}^{n-1}$.
• Decomposition of $u$ into radial and angular parts via spherical harmonics, reducing the problem to a one-dimensional ODE on $r = |x|$.
• Derivation of the best constant $\mu_N(\Omega;\alpha)$ in the Navier case via minimization over $k \in \mathbb{N} \cup \{0\}$: $\mu_{n,\alpha} = \min_k |\gamma_{n,\alpha} + k(n-2+k)|^2$, where $\gamma_{n,\alpha} = \left(\frac{n-2}{2}\right)^2 - \left(\frac{\alpha-2}{2}\right)^2$.
• Construction of test functions $u(x) = |x|^{(4-n-\alpha)/2} v(-\log|x|) \varphi(\sigma)$, where $\varphi$ is an eigenfunction of $-\Delta_\sigma$ with eigenvalue $\lambda_\Sigma$, to analyze the sharpness of remainder terms.
• Application of scaling arguments ($t \to 0$ and $t \to \infty$) on rescaled test functions to derive upper bounds on the constants $A$ and $B$ in remainder terms of the form $A \int |x|^{\alpha-4} |\log|x||^{-2} |u|^2 + B \int |x|^{\alpha-4} |\log|x||^{-4} |u|^2$.
• Proof of sharpness by contradiction: assuming larger constants leads to violation of known optimal inequalities on the half-line $\mathbb{R}_+$.

Experimental results

Research questions

• RQ1 What is the best constant $\mu_N(\Omega;\alpha)$ in the weighted Rellich inequality for $u \in C^2_c(\overline{\Omega} \setminus \{0\})$ on a cone $\Omega = \mathcal{C}_\Sigma$?
• RQ2 How does the best constant $\mu_D(\Omega;\alpha)$ in the Dirichlet case compare to $\mu_N(\Omega;\alpha)$, and what is its lower bound?
• RQ3 Under what conditions does the weighted Rellich inequality fail, and how is this related to the spectrum of the Laplace-Beltrami operator?
• RQ4 What are the optimal constants in improved Rellich inequalities with logarithmic remainder terms on bounded or exterior cone-like domains?
• RQ5 Can the sharpness of the logarithmic remainder terms be rigorously proven using scaling and variational techniques?

Key findings

• The best constant in the Navier case on $\Omega = \mathcal{C}_\Sigma$ is $\mu_N(\mathcal{C}_\Sigma;\alpha) = \min_{k \in \mathbb{N} \cup \{0\}} |\gamma_{n,\alpha} + k(n-2+k)|^2$, where $\gamma_{n,\alpha} = \left(\frac{n-2}{2}\right)^2 - \left(\frac{\alpha-2}{2}\right)^2$.
• The inequality fails if and only if $-\gamma_{n,\alpha}$ is an eigenvalue of $-\Delta_\sigma$ on $\mathbb{S}^{n-1}$, revealing a resonance mechanism.
• For $n=2$, the optimal constant in the Navier case on $\mathbb{B}^2 \setminus \{0\}$ or $\mathbb{R}^2 \setminus \overline{\mathbb{B}^2}$ is $\mu_{2,\alpha} = \frac{1}{2}$, with sharp remainder $\frac{1}{2} \int |x|^{-4} |\log|x||^{-2} |u|^2$.
• For $n \geq 3$, the sharp remainder in the Navier case on $\mathbb{B}^n \setminus \{0\}$ or $\mathbb{R}^n \setminus \overline{\mathbb{B}^n}$ is $\frac{n^2 - 4n + 8}{8} \int |x|^{-4} |\log|x||^{-2} |u|^2$.
• In the Dirichlet case, the sharp remainder includes an additional term: $\frac{9}{16} \int |x|^{-4} |\log|x||^{-4} |u|^2$, which is optimal as shown via scaling limits.
• The constants $A = \frac{\overline{\gamma}_{n,\alpha} + \lambda_\Sigma}{2}$ and $B = \left(\frac{3}{4}\right)^2 = \frac{9}{16}$ in the remainder terms are sharp, as proven by testing on scaled functions and taking limits $t \to 0$ and $t \to \infty$.