On the Structure of Periodic Eigenvalues of the Vectorial $p$-Laplacian

TL;DR

This paper solves a long-standing open problem by characterizing the structure of periodic eigenvalues of the vectorial p-Laplacian, revealing infinitely many eigenvalue sequences for p ≠ 2 using integrability and number theory.

Contribution

It provides a complete description of the eigenvalue structure for the vectorial p-Laplacian, introducing the notion of scaling momenta and leveraging Hamiltonian system integrability.

Findings

Infinitely many eigenvalue sequences exist for p ≠ 2.

Eigenvalues are constructed via scaling momenta.

Numerical simulations illustrate the new eigenvalue sequences.

Abstract

In this paper we will solve an open problem raised by Man\'asevich and Mawhin twenty years ago on the structure of the periodic eigenvalues of the vectorial $p$-Laplacian. This is an Euler-Lagrangian equation on the plane or in higher dimensional Euclidean spaces. The main result obtained is that for any exponent $p$ other than $2$, the vectorial $p$-Laplacian on the plane will admit infinitely many different sequences of periodic eigenvalues with a given period. These sequences of eigenvalues are constructed using the notion of scaling momenta we will introduce. The whole proof is based on the complete integrability of the equivalent Hamiltonian system, the tricky reduction to $2$-dimensional dynamical systems, and a number-theoretical distinguishing between different sequences of eigenvalues. Some numerical simulations to the new sequences of eigenvalues and eigenfunctions will be given. Several further conjectures towards to the panorama of the spectral sets will be imposed.