[Paper Review] Hardy's inequality in a limiting case on general bounded domains

Jaeyoung Byeon, Futoshi Takahashi|arXiv (Cornell University)|Jul 13, 2017

Nonlinear Partial Differential Equations15 references3 citations

TL;DR

This paper investigates the best constant $ C_N(\Omega) $ in a limiting case of Hardy's inequality on bounded domains in $ \mathbb{R}^N $, where the weight involves $ |x|^{-N} \left(\log \frac{R}{|x|}\right)^{-N} $. It establishes sufficient conditions for $ C_N(\Omega) > C_N(B_R) $ and proves that $ C_N(\Omega) $ is not attained for certain domains, including a specific planar domain in $ \mathbb{R}^2 $ with $ C_2(\Omega) > 1/4 $.

ABSTRACT

In this paper, we study Hardy's inequality in a limiting case: $$ \int_Ω | abla u |^N dx \ge C_N(Ω) \int_Ω \frac{|u(x)|^N}{|x|^N \left(\log \frac{R}{|x|} ight)^N} dx $$ for functions $u \in W^{1,N}_0(Ω)$, where $Ω$ is a bounded domain in $\mathbb{R}^N$ with $R = \sup_{x \in Ω} |x|$. We study the (non-)attainability of the best constant $C_N(Ω)$ in several cases. We provide sufficient conditions that assure $C_N(Ω) > C_N(B_R)$ and $C_N(Ω)$ is attained, here $B_R$ is the $N$-dimensional ball with center the origin and radius $R$. Also we provide an example of $Ω\subset \mathbb{R}^2$ such that $C_2(Ω) > C_2(B_R) = 1/4$ and $C_2(Ω)$ is not attained.

Motivation & Objective

• To analyze the best constant $ C_N(\Omega) $ in the limiting case of Hardy's inequality on general bounded domains in $ \mathbb{R}^N $.
• To determine conditions under which $ C_N(\Omega) > C_N(B_R) $, where $ B_R $ is the ball of radius $ R $ centered at the origin.
• To investigate the attainability of the best constant $ C_N(\Omega) $ in the critical $ L^N $-case.
• To construct an explicit example in $ \mathbb{R}^2 $ where $ C_2(\Omega) > 1/4 $ and $ C_2(\Omega) $ is not attained.

Proposed method

• Define $ C_N(\Omega) $ as the infimum of the ratio $ \frac{\int_\Omega |\nabla u|^N dx}{\int_\Omega \frac{|u|^N}{|x|^N (\log \frac{R}{|x|})^N} dx} $ over non-zero functions in $ W^{1,N}_0(\Omega) $.
• Use a change of variables and spherical coordinates to reduce the problem to a one-dimensional problem involving angular and radial derivatives.
• Apply a variational method to estimate the best constant via a minimization problem over functions on $ (0, \pi) $, leading to a Sturm-Liouville-type eigenvalue problem.
• Construct a domain $ \Omega \subset \mathbb{R}^2 $ with a specific angular behavior near the boundary to ensure $ C_2(\Omega) > 1/4 $.
• Use integration by parts and test functions of the form $ u_\alpha(\theta) = (\sin \theta)^\alpha $ to estimate the infimum and show it approaches $ 1/4 $ from above.
• Prove non-attainability by showing that any minimizer would require radial derivative to vanish, contradicting the zero boundary condition.

Experimental results

Research questions

• RQ1 Under what conditions is the best constant $ C_N(\Omega) $ in the limiting Hardy inequality strictly greater than $ C_N(B_R) $?
• RQ2 When is the best constant $ C_N(\Omega) $ attained in $ W^{1,N}_0(\Omega) $?
• RQ3 Can a domain be constructed such that $ C_N(\Omega) > C_N(B_R) $ and $ C_N(\Omega) $ is not attained?
• RQ4 What is the role of the geometry of the domain near the origin and the boundary in determining the sharpness and attainability of the constant?

Key findings

• For a specific domain $ \Omega \subset \mathbb{R}^2 $, the best constant satisfies $ C_2(\Omega) > 1/4 = C_2(B_R) $, showing strict improvement over the ball.
• The best constant $ C_N(\Omega) $ is not attained in $ W^{1,N}_0(\Omega) $ for this constructed domain, despite being strictly greater than $ C_N(B_R) $.
• The infimum $ E = \inf \frac{\int_0^\pi (\phi_\theta)^2 d\theta}{\int_0^\pi \frac{\phi^2}{\sin^2 \theta} d\theta} $ equals $ 1/4 $, but is not achieved in $ W^{1,2}_0(0,\pi) $.
• The constant $ C_N(\Omega) $ is bounded below by $ \left(\frac{N-1}{N}\right)^N $, with equality only in the limit for the ball $ B_R $.
• The scaling invariance of the inequality under $ u_\lambda(x) = \lambda^{-(N-1)/N} u\left( (|x|/R)^{\lambda-1} x \right) $ plays a key role in the analysis.
• Non-attainability arises because any minimizer would need to be radial, but such functions cannot satisfy the zero trace condition unless trivial.