Hardy type inequalities in higher dimensions with explicit estimate of constants

F. G. Avkhadiev

Lobachevskii journal of mathematics, Tome 21 (2006), pp. 3-31

Résumé

Let $\Omega$ be an open set in $\mathbb R^n$ such that $\Omega\ne\mathbb R^n$. For $1\le p\infty$, $1$ and $\delta=\operatorname{dist}(x,\partial\Omega)$ we estimate the Hardy constant $$ c_p(s,\Omega)=\sup\{\|f/\delta^{s/p}\|_{L^p(\Omega)}:f\in C_0^\infty(\Omega),\ \|(\nabla f)/\delta^{s/p-1}\|_{L^p(\Omega)}=1\} $$ and some related quantities. For open sets $\Omega\subset\mathbb R^2$ we prove the following bilateral estimates $$ \min\{2,p\}M_0(\Omega)\le c_p(2,\Omega)\le 2p(\pi M_0(\Omega)+a_0)^2, \quad a_0=4.38, $$ where $M_0(\Omega)$ is the geometrical parameter defined as the maximum modulus of ring domains in $\Omega$ with center on $\partial\Omega$. Since the condition $M_0 (\Omega)\infty$ means the uniformly perfectness of $\partial\Omega$, these estimates give a direct proof of the following Ancona–Pommerenke theorem: $c_2(2,\Omega)$ is finite if and only if the boundary set $\partial\Omega$ is uniformly perfect (see [2], [22] and [40]). Moreover, we obtain the following direct extension of the one dimensional Hardy inequality to the case $n\ge 2$: if $s>n$, then for arbitrary open sets $\Omega\subset\mathbb R^n$ ($\Omega\ne\mathbb R^n$) and any $p\in[1,\infty)$ the sharp inequality $c_p(s,\Omega)\le p/(s-n)$ is valid. This gives a solution of a known problem due to J. L. Lewis [31] and A. Wannebo [44]. Estimates of constants in certain other Hardy and Rellich type inequalities are also considered. In particular, we obtain an improved version of a Hardy type inequality by H. Brezis and M. Marcus [13] for convex domains and give its generalizations.

Détail

Keywords: Hardy type inequalities, distance to the boundary, uniformly perfect sets, Rellich type inequalities.