The Integrability of Negative Powers of the Solution of the Saint Venant Problem

Abstract

We initiate the study of the finiteness condition ${\int }_{\Omega }u(x{)}^{-\beta }dx\le C(\Omega ,\beta )<+\infty$ where $\Omega \subseteq {\mathbb{R}}^n$ is an open set and $u$ is the solution of the Saint Venant problem $\Delta u=-1$ in $\Omega$, $u=0$ on $\partial \Omega$. The central issue which we address is that of determining the range of values of the parameter $\beta >0$ for which the aforementioned condition holds under various hypotheses on the smoothness of $\Omega$ and demands on the nature of the constant $C(\Omega ,\beta )$. Classes of domains for which our analysis applies include bounded piecewise $C^1$ domains in ${\mathbb{R}}^n$, $n\ge 2$, with conical singularities (in particular polygonal domains in the plane), polyhedra in ${\mathbb{R}}^3$, and bounded domains which are locally of class $C^2$ and which have (finitely many) outwardly pointing cusps. For example, we show that if $u_N$ is the solution of the Saint Venant problem in the regular polygon ${\Omega }_N$ with $N$ sides circumscribed by the unit disc in the plane, then for each $\beta \in (0,1)$ the following asymptotic formula holds: % {eqnarray*} \int_{Ω_N}u_N(x)^{-β}\,dx=\frac{4^βπ}{1-β} +{\mathcal{O}}(N^{β-1})\quad{as}\,\,N\to\infty. {eqnarray*} % One of the original motivations for addressing the aforementioned issues was the study of sublevel set estimates for functions $v$ satisfying $v(0)=0$, $\nabla v(0)=0$ and $\Delta v\ge c>0$.

To appear, Annali Della Scuola Normale Superiore, Part 2, Vol. XIII, series V, June 2014