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

We initiate the study of the finiteness condition $\int_Ωu(x)^{-β}\,dx\leq C(Ω,β)<+\infty$ where $Ω\subseteq{\mathbb{R}}^n$ is an open set and $u$ is the solution of the Saint Venant problem $Δu=-1$ in $Ω$, $u=0$ on $\partialΩ$. The central issue which we address is that of determining the range of values of the parameter $β>0$ for which the aforementioned condition holds under various hypotheses on the smoothness of $Ω$ and demands on the nature of the constant $C(Ω,β)$. 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 $Ω_N$ with $N$ sides circumscribed by the unit disc in the plane, then for each $β\in(0,1)$ 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 $Δv\geq c>0$.

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