Hardy Constants on Wedged Domains

Abstract

The Hardy constant for a domain Ω ⊂ ℝⁿ is defined as the best constant for the Hardy inequality

\begin{array}{c}{\int }_{\mathrm{\Omega }}{\mid \mathrm{\nabla }u\mid }^{2}dx\ge C{\int }_{\mathrm{\Omega }}\frac{{\mid u\mid }^2}{{\mid x\mid }^2}dx,u\in C_0^{\mathrm{\infty }}(\mathrm{\Omega }).\end{array}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\matrix{{\int_\Omega {{{\left| {\nabla u} \right|}^2}dx \ge C\int_\Omega {{{{{\left| u \right|}^2}} \over {{{\left| x \right|}^2}}}dx,}}}\quad{u \in C_0^\infty (\Omega).}}$$\end{document}

In this paper we determine the Hardy constants for wedged domains. In particular we show the dependence of Hardy constant on n and the angle φ of wedged domain. Our results also reflect the influence of boundary smoothness and location of singular point on Hardy constant. Some improved Hardy inequalities on special wedged domains are also obtained.

Keywords

Hardy constant / wedged domain / improved Hardy inequality / 42B37 / 35A23

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Peking University