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Finsler $p$-Laplacian in domains becoming unbounded

Luca Esposito, Lorenzo Lamberti, Dattatreya N. N., Prosenjit Roy

TL;DR

The paper analyzes the asymptotic behavior of solutions, energies, and the first eigenvalue for the anisotropic Finsler $p$-Laplacian on growing cylinders $\\Omega_\ell=\\ell \\omega_1 imes \\omega_2$ as $\\ell\to\infty$. It proves that solutions converge to the cross-section solution $u_ fty$ with a polynomial rate in general, and identifies exponential convergence in a restricted class of $H$; energy functionals per cross-section converge with an $O(1/\\ell)$ correction; and the first eigenvalue $\\lambda_\ell^1$ converges to the cross-sectional eigenvalue $\\mu_\infty$ with explicit rates in a subclass. The results unify and extend known convergence phenomena from Laplacian and $p$-Laplacian settings to the Finsler (anisotropic) framework, employing variational methods, monotonicity, Picone-type identities, and hole-filling arguments. The findings have implications for understanding dimensional reduction in anisotropic media and for spectral asymptotics of nonlinear elliptic operators in unbounded cylindrical domains.

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.

Finsler $p$-Laplacian in domains becoming unbounded

TL;DR

The paper analyzes the asymptotic behavior of solutions, energies, and the first eigenvalue for the anisotropic Finsler -Laplacian on growing cylinders as . It proves that solutions converge to the cross-section solution with a polynomial rate in general, and identifies exponential convergence in a restricted class of ; energy functionals per cross-section converge with an correction; and the first eigenvalue converges to the cross-sectional eigenvalue with explicit rates in a subclass. The results unify and extend known convergence phenomena from Laplacian and -Laplacian settings to the Finsler (anisotropic) framework, employing variational methods, monotonicity, Picone-type identities, and hole-filling arguments. The findings have implications for understanding dimensional reduction in anisotropic media and for spectral asymptotics of nonlinear elliptic operators in unbounded cylindrical domains.

Abstract

We study the asymptotic behavior of sequences of solutions, energies functionals, and the first eigenvalues associated with the Finsler -Laplace operator, also known as the anisotropic -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.

Paper Structure

This paper contains 11 sections, 15 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Notation
  4. Asymptotics of solutions
  5. Proof of the Theorem \ref{['thm poly convergence']}
  6. Exponential rate of convergence
  7. Energy asymptotics
  8. An eigenvalue problem
  9. Proof of Theorem \ref{['eigenvalue from above']}
  10. Exact rate of convergence
  11. Appendix

Key Result

Theorem 1.4

Let $u_{\infty}$ be the solution of For $p\geq 2$, it holds that And For $1<p<2$, it holds that where, $C_{i}$, for $~i=1,2$, is a positive constant and $u_{\infty}$, originally defined on $\omega_2$, is to be understood as extended constantly with respect to the variables $X_1$.

Theorems & Definitions (34)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 24 more