[Paper Review] On the Hardy-Poincaré inequality with boundary singularities

Mouhamed Moustapha Fall|arXiv (Cornell University)|Aug 27, 2010

Nonlinear Partial Differential Equations9 references4 citations

TL;DR

This paper establishes the existence of a finite critical threshold $\lambda^*\in\mathbb{R}$ for the Hardy-Poincar{\'e} inequality with a boundary-singular weight $|x|^{-2}$ in smooth bounded domains $\Omega\subset\mathbb{R}^N$, $N\geq2$, showing that minimizers exist if and only if $\lambda > \lambda^*$. The key result resolves an open question by proving $\lambda^*$ is always finite and that no minimizer exists at $\lambda^*$, even for domains with negative curvature at the origin.

ABSTRACT

Let $Ø$ be a smooth bounded domain in $\R^N$ with $N\ge 1$. In this paper we study the Hardy-Poincaré inequality with weight function singular at the boundary of $Ø$. In particular we provide sufficient and necessary conditions on the existence of minimizers.

Motivation & Objective

• To resolve the open problem of whether the critical threshold $\lambda^*$ is always finite for smooth bounded domains with boundary singularity at $0\in\partial\Omega$.
• To determine the precise conditions under which minimizers exist or fail to exist for the weighted Hardy-Poincar{\'e} quotient with $|x|^{-2}$ weight.
• To establish that $\lambda^*\in\mathbb{R}$ and that no extremal exists at $\lambda^*$, even in domains with negative principal curvature at the origin.
• To generalize the analysis to weighted variational problems with singular weights depending on distance to submanifolds of $\partial\Omega$.

Proposed method

• Uses Fermi coordinates near the boundary to linearize the geometry and express the Laplacian in normal coordinates with curvature corrections.
• Applies a refined local Hardy inequality (1.3) with improved constants to control the singularity near $\partial\Omega$, proving $\lambda^*\in\mathbb{R}$.
• Employs a maximum principle argument for supersolutions of the equation $-\Delta u - \frac{N^2}{4}|x|^{-2}u = \lambda u$ to show non-existence of nontrivial nonnegative supersolutions in $H^1_0(\Omega)\cap C(\Omega)$ at $\lambda^*$.
• Constructs explicit barrier functions $w_a$ with power-like decay to prove that any nontrivial supersolution would violate $L^2$ integrability of $u/|x|$, leading to contradiction.
• Uses variational methods and compactness arguments to prove existence of minimizers for $\lambda > \lambda^*$, relying on the strict inequality $J_\lambda < \frac{N^2}{4}$.
• Generalizes results to higher-codimensional submanifolds $\Sigma_k\subset\partial\Omega$ by introducing weighted quotients with $\textrm{dist}(x,\Sigma_k)^{-2}$ and analyzing critical thresholds via geometric integrals $I_k$.

Experimental results

Research questions

• RQ1 Is the critical threshold $\lambda^*$ always finite for smooth bounded domains $\Omega\subset\mathbb{R}^N$, $N\geq2$, with $0\in\partial\Omega$?
• RQ2 Does the Hardy-Poincar{\'e} quotient with $|x|^{-2}$ weight admit a minimizer at $\lambda^*$, even when the domain has negative curvature at the origin?
• RQ3 Can the non-existence of extremals at $\lambda^*$ be proven via maximum principle and barrier function techniques?
• RQ4 How does the existence of minimizers depend on the global geometry of $\Omega$ rather than just local curvature at $0$?
• RQ5 Can the results be generalized to singular weights based on distance to submanifolds $\Sigma_k\subset\partial\Omega$ of codimension $k$?

Key findings

• The critical threshold $\lambda^*(\Omega)$ is always finite and belongs to $\mathbb{R}$ for any smooth bounded domain $\Omega\subset\mathbb{R}^N$, $N\geq2$, with $0\in\partial\Omega$.
• Minimizers for the Hardy-Poincar{\'e} quotient exist if and only if $\lambda > \lambda^*(\Omega)$, resolving an open question from previous works.
• No nontrivial nonnegative supersolution exists in $H^1_0(\Omega)\cap C(\Omega)$ for the equation $-\Delta u - \frac{N^2}{4}|x|^{-2}u = \lambda^* u$, implying no extremal exists at $\lambda^*$.
• The supremum of the quotient $\mu_\lambda(\Omega)$ is exactly $\frac{N^2}{4}$, achieved in the limit as $\lambda\to\lambda^{*+}$, and this value equals the Hardy constant of the half-space $\mathbb{R}^N_+$.
• For domains with negative principal curvature at $0$, the Hardy constant remains $\frac{N^2}{4}$, showing that the existence of minimizers depends on global geometry, not just local curvature.
• The results generalize to $k$-dimensional submanifolds $\Sigma_k\subset\partial\Omega$, where the critical threshold for the quotient with $\textrm{dist}(x,\Sigma_k)^{-2}$ weight is finite and extremals exist iff $\lambda > \lambda^*$, with $\lambda^*$ finite if and only if $I_k < \infty$.