Calculus of Variations and Geometric Measure Theory

M. van den Berg - T. Kappeler

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

created by vandenberg1 on 27 Feb 2018


Submitted Paper

Inserted: 27 feb 2018

Year: 2018

ArXiv: 1802.05499 PDF


Bounds are obtained for the $L^p$ norm of the torsion function $v_{\Omega}$, i.e. the solution of $-\Delta v=1,\, v\in H_0^1(\Omega),$ in terms of the Lebesgue measure of $\Omega$ and the principal eigenvalue $\lambda_1(\Omega)$ of the Dirichlet Laplacian acting in $L^2(\Omega)$. We show that these bounds are sharp for $1\le p\le 2$.