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

Michiel van den Berg; Thomas Kappeler

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

Analysis of PDEs 2018-02-16 v1

Authors: Michiel van den Berg , Thomas Kappeler

Abstract

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 $1\le p\le 2$ .

Keywords

torsion function lp norm dirichlet laplacian principal eigenvalue sharp bounds

Cite

@article{arxiv.1802.05499,
  title  = {On the $L^p$ norm of the torsion unction},
  author = {Michiel van den Berg and Thomas Kappeler},
  journal= {arXiv preprint arXiv:1802.05499},
  year   = {2018}
}

Comments

14 pages

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