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

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

2012-04-20

arxiv.org/abs/1204.4590v1

Mathematics

Analysis of PDEs

Scientific paper

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 $\Delta v\geq c>0$.

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