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

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 $u_t-\Delta_p^Nu=0$.