Differential and Integral Equations

Radial eigenfunctions for the game-theoretic $p$-Laplacian on a ball

Bernd Kawohl, Stefan Krӧmer, and Jannis Kurtz

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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 $u_t-\Delta_p^Nu=0$.

Article information

Source
Differential Integral Equations, Volume 27, Number 7/8 (2014), 659-670.

Dates
First available in Project Euclid: 6 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.die/1399395747

Mathematical Reviews number (MathSciNet)
MR3200758

Zentralblatt MATH identifier
1340.35242

Subjects
Primary: 35B25: Singular perturbations 35Q91: PDEs in connection with game theory, economics, social and behavioral sciences 35J60: Nonlinear elliptic equations 35J70: Degenerate elliptic equations 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems

Citation

Kawohl, Bernd; Krӧmer, Stefan; Kurtz, Jannis. Radial eigenfunctions for the game-theoretic $p$-Laplacian on a ball. Differential Integral Equations 27 (2014), no. 7/8, 659--670. https://projecteuclid.org/euclid.die/1399395747


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