Vol. 8, No. 5, 2015

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A numerical investigation of level sets of extremal Sobolev functions

Stefan Juhnke and Jesse Ratzkin

Vol. 8 (2015), No. 5, 787–799
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.

Keywords
extremal Sobolev functions, semilinear elliptic PDE, distribution function
Primary: 65N30
Secondary: 35J20