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 Ω n and 1 p < 2n(n 2), these functions extremize the ratio uL2(Ω)uLp(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