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July/August 2014 Radial eigenfunctions for the game-theoretic $p$-Laplacian on a ball
Bernd Kawohl, Stefan Krӧmer, Jannis Kurtz
Differential Integral Equations 27(7/8): 659-670 (July/August 2014).

Abstract

The normalized or game-theoretic $p$-Laplacian operator given by $$ -\Delta_p^Nu:=-\frac{1}{p}|\nabla u|^{2-p}\Delta_p(u) $$ for $p\in(1,\infty)$ with $\Delta_pu=\rm{div}(|\nabla u|^{p-2}\nabla u)$ has no apparent variational structure. Showing the existence of a first (positive) eigenvalue of this fully nonlinear operator requires heavy machinery as in [6]. If it is restricted to the class of radial functions, however, the normalized $p$-Laplacian transforms into a linear Sturm-Liouville operator. We investigate radial eigenfunctions to this operator under homogeneous Dirichlet boundary conditions and come up with an explicit complete orthonormal system of Bessel functions in a suitably weighted $L^2$-space. This allows us to give a Fourier-series representation for radial solutions to the corresponding evolution equation $u_t-\Delta_p^Nu=0$.

Citation

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Bernd Kawohl. Stefan Krӧmer. Jannis Kurtz. "Radial eigenfunctions for the game-theoretic $p$-Laplacian on a ball." Differential Integral Equations 27 (7/8) 659 - 670, July/August 2014.

Information

Published: July/August 2014
First available in Project Euclid: 6 May 2014

zbMATH: 1340.35242
MathSciNet: MR3200758

Subjects:
Primary: 35B25, 35J60, 35J70, 35Q91, 46E35

Rights: Copyright © 2014 Khayyam Publishing, Inc.

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Vol.27 • No. 7/8 • July/August 2014
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