Finsler $p$-Laplacian in domains becoming unbounded

TL;DR

This paper investigates the asymptotic behavior of solutions, energies, and eigenvalues of the Finsler $p$-Laplace operator on elongated domains, revealing convergence patterns and rates as the domain length tends to infinity.

Contribution

It establishes convergence results and rates for solutions, energies, and eigenvalues of the Finsler $p$-Laplace operator on unbounded domains, extending understanding of anisotropic $p$-Laplace problems.

Findings

Solutions converge to the cross-section solution with polynomial or exponential rate.

Energies on finite cylinders converge to the cross-section energy with scaling.

First eigenvalues converge, with optimal rate in specific cases.

Abstract

We study the asymptotic behavior of sequences of solutions, energies functionals, and the first eigenvalues associated with the Finsler $p$-Laplace operator, also known as the anisotropic $p$-Laplace operator on a sequence of bounded cylinders whose length tends to infinity. We prove that the solutions on the bounded cylinders converge to the solution on the cross-section, with a polynomial rate of convergence in the general case and exponential convergence in some special cases. We show that energies on finite cylinders, with the multiplication of a scaling factor, converge to the energy on the cross-section. Finally, we investigate the convergence of the first eigenvalue and, for a specific subclass, we provide the optimal convergence rate.