Hardy's inequality in a limiting case on general bounded domains

Abstract

In this paper, we study Hardy's inequality in a limiting case: $${\int }_{\Omega }|\nabla u{|}^{N}dx\ge C_N(\Omega ){\int }_{\Omega }\frac{|u(x){|}^N}{|x{|}^N{\left(\log \frac{R}{|x|}\right)}^N}dx$$ for functions $u\in W_0^{1,N}(\Omega )$, where $\Omega$ is a bounded domain in ${\mathbb{R}}^N$ with $R={\sup }_{x\in \Omega }|x|$. We study the (non-)attainability of the best constant $C_N(\Omega )$ in several cases. We provide sufficient conditions that assure $C_N(\Omega )>C_N(B_R)$ and $C_N(\Omega )$ is attained, here $B_R$ is the $N$-dimensional ball with center the origin and radius $R$. Also we provide an example of $\Omega \subset {\mathbb{R}}^2$ such that $C_2(\Omega )>C_2(B_R)=1/4$ and $C_2(\Omega )$ is not attained.

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