Bivariate Hardy-Sobolev Inequality and Its Sharp Stability

TL;DR

This paper proves a new bivariate Hardy-Sobolev inequality, finds the optimal constant, characterizes minimizers, and establishes sharp stability results for nonnegative functions in the entire space.

Contribution

It introduces a novel bivariate Hardy-Sobolev inequality with explicit best constants and stability analysis, extending previous univariate results.

Findings

Established the explicit best constant for the inequality.

Characterized the set of minimizers.

Proved sharp stability results for nonnegative functions.

Abstract

This paper establishes a bivariate Hardy-Sobolev inequality. Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be an open domain, $s \in (0,2)$, $\alpha > 1$, $\beta > 1$ with $\alpha + \beta = 2^*(s)$, and $\kappa \in \mathbb{R}$. For any functions $u, v \in D_0^{1,2}(\Omega)$, we prove the inequality: \begin{multline*} \int_{\Omega} |\nabla u|^2 \, \mathrm{d}x + \int_{\Omega} |\nabla v|^2 \, \mathrm{d}x \ge S_{\alpha,\beta,\lambda,\mu}(\Omega) \left( \int_{\Omega} \Big( \lambda \frac{|u|^{2^*(s)}}{|x|^s} + \mu \frac{|v|^{2^*(s)}}{|x|^s} + 2^*(s) \kappa \frac{|u|^\alpha |v|^\beta}{|x|^s} \Big)\, \mathrm{d}x \right)^{\frac{2}{2^*(s)}}. \end{multline*} We derive the best constant $S_{\alpha,\beta,\lambda,\mu}(\Omega)$ and characterize the set of minimizers. Moreover, for $\Omega = \mathbb{R}^N$ and $\kappa > 0$, we obtain sharp stability results for nonnegative functions.