The Neumann eigenvalue problem for the $\infty$-Laplacian

TL;DR

This paper investigates the behavior of the first nontrivial eigenfunction of the Neumann $p$-Laplacian as $p$ approaches infinity, showing convergence to a viscosity solution of the $ abla$-Laplacian eigenproblem and establishing related inequalities.

Contribution

It demonstrates the convergence of eigenfunctions and eigenvalues for the Neumann $p$-Laplacian to the $ abla$-Laplacian case, providing new nonlinear analogues of classical inequalities.

Findings

Eigenfunctions converge to viscosity solutions of the $ abla$-Laplacian eigenproblem.

Eigenvalues converge to the first nonzero eigenvalue for convex sets.

Derived nonlinear inequalities analogous to classical linear results.

Abstract

The first nontrivial eigenfunction of the Neumann eigenvalue problem for the $p$-Laplacian, suitable normalized, converges as $p$ goes to $\infty$ to a viscosity solution of an eigenvalue problem for the $\infty$-Laplacian. We show among other things that the limit of the eigenvalue, at least for convex sets, is in fact the first nonzero eigenvalue of the limiting problem. We then derive a number of consequences, which are nonlinear analogues of well-known inequalities for the linear (2-)Laplacian.