Bivariate Hardy-Sobolev Inequality and Its Sharp Stability
Yingfang Zhang, Xuexiu Zhong, Wenming Zou
TL;DR
The paper develops a sharp, variational framework for a bivariate Hardy-Sobolev inequality on domains Ω ⊂ ℝ^N, establishing the best constant S_{α,β,λ,μ}(Ω) and a precise characterization of minimizers. For κ ≤ 0, the inequality reduces to a semitrivial combination of single-variable Hardy-Sobolev terms, while κ > 0 leads to a reduced problem involving a scalar t and the function g(t), yielding explicit forms for minimizers when Ω = ℝ^N. The authors prove a sharp stability estimate around the minimizers in ℝ^N, with the deficit controlling the gradient distance to the minimizers via an exponent ι that is 1 in nondegenerate cases and 1/2 in degenerate ones, and they provide a weighted extension through a suitable transformation. These results integrate spectral analysis, a detailed study of g(t), and a careful nonlinear analysis to deliver both the best constants and quantitative stability for the bivariate Hardy-Sobolev inequality, significantly advancing the theory beyond the univariate case.
Abstract
This paper establishes a bivariate Hardy-Sobolev inequality. Let $Ω\subset \mathbb{R}^N$ ($N \geq 3$) be an open domain, $s \in (0,2)$, $α> 1$, $β> 1$ with $α+ β= 2^*(s)$, and $κ\in \mathbb{R}$. For any functions $u, v \in D_0^{1,2}(Ω)$, we prove the inequality: \begin{multline*} \int_Ω |\nabla u|^2 \, \mathrm{d}x + \int_Ω |\nabla v|^2 \, \mathrm{d}x \ge S_{α,β,λ,μ}(Ω) \left( \int_Ω \Big( λ\frac{|u|^{2^*(s)}}{|x|^s} + μ\frac{|v|^{2^*(s)}}{|x|^s} + 2^*(s) κ\frac{|u|^α|v|^β}{|x|^s} \Big)\, \mathrm{d}x \right)^{\frac{2}{2^*(s)}}. \end{multline*} We derive the best constant $S_{α,β,λ,μ}(Ω)$ and characterize the set of minimizers. Moreover, for $Ω= \mathbb{R}^N$ and $κ> 0$, we obtain sharp stability results for nonnegative functions.