Hardy's inequality in a limiting case on general bounded domains
In this paper, we study Hardy's inequality in a limiting case: $$ \int_Ω |\nabla u |^N dx \ge C_N(Ω) \int_Ω \frac{|u(x)|^N}{|x|^N \left(\log \frac{R}{|x|} \right)^N} dx $$ 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.
math.AP↗