| Abstract: |
| 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 \( \log \frac{R}{|x|} \)^N} dx$
for functions $u \in W^{1,N}_0(\Omega)$, where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with $R = \sup_{x \in \Omega} |x|$.
We study the 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 $\Omega \subset \re^2$ such that $C_2(\Omega) > C_2(B_R) = 1/4$ and $C_2(\Omega)$ is not attained. |
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