On the $L^p$ norm of the torsion unction

Bounds are obtained for the $L^p$ norm of the torsion function $v_{\mathrm{\Omega }}$, i.e. the solution of $-\mathrm{\Delta }v=1,v\in H_0^1(\mathrm{\Omega }),$ in terms of the Lebesgue measure of $\mathrm{\Omega }$ and the principal eigenvalue ${\lambda }_1(\mathrm{\Omega })$ of the Dirichlet Laplacian acting in $L^2(\mathrm{\Omega })$. We show that these bounds are sharp for $1\le p\le 2$.