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

We initiate the study of the finiteness condition $\int_{\Omega}u(x)^{-\beta}\,dx\leq C(\Omega,\beta)<+\infty$ 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 ${\mathbb{R}}^n$, $n\geq 2$, 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_{\Omega_N}u_N(x)^{-\beta}\,dx=\frac{4^\beta\pi}{1-\beta} +{\mathcal{O}}(N^{\beta-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 $\Delta v\geq c>0$.