A numerical investigation of level sets of extremal Sobolev functions

Abstract

We investigate the level sets of extremal Sobolev functions. For $\Omega \subset {ℝ}^n$ and $1\le p<2n∕\left(n-2\right)$, these functions extremize the ratio $\parallel \nabla u{\parallel }_{L^2\left(\Omega \right)}∕\parallel u{\parallel }_{L^p\left(\Omega \right)}$. We conjecture that as $p$ increases, the extremal functions become more “peaked” (see the introduction below for a more precise statement), and present some numerical evidence to support this conjecture.