  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 $\Delta v\geq c>0$.
