A numerical investigation of level sets of extremal Sobolev functions

Juhnke, Stefan; Ratzkin, Jesse

doi:10.2140/involve.2015.8.787

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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

DOI: 10.2140/involve.2015.8.787

Abstract

We investigate the level sets of extremal Sobolev functions. For $Ω \subset ℝ^{n}$ and $1 \leq p < 2 n ∕ (n - 2)$ , these functions extremize the ratio $∥ \nabla u ∥_{L^{2} (Ω)} ∕ ∥ u ∥_{L^{p} (Ω)}$ . 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

Mathematical Subject Classification 2010

Primary: 65N30

Secondary: 35J20

Milestones

Received: 25 March 2014

Revised: 7 September 2014

Accepted: 1 October 2014

Published: 28 September 2015

Communicated by Kenneth S. Berenhaut

Authors

Stefan Juhnke
Department of Mathematics and Applied Mathematics University of Cape Town, Private Bag X1 Rondebosch Cape Town 7701 South Africa

Jesse Ratzkin
Department of Mathematics and Applied Mathematics University of Cape Town, Private Bag X1 Rondebosch Cape Town 7701 South Africa

Vol. 8, No. 5, 2015