[Paper Review] Fractional eigenvalue problems that approximate Steklov eigenvalues

Leandro M. Del Pezzo, Julio D. Rossi+1 more|arXiv (Cornell University)|Jan 20, 2016

Nonlinear Partial Differential Equations14 references3 citations

TL;DR

This paper introduces a fractional eigenvalue problem involving the nonlocal p-Laplacian and a boundary-weighted term that approximates the classical Steklov eigenvalue problem as the fractional parameter $ s \to 1^{-} $. The key result establishes that the first eigenvalue of the fractional problem converges to the first Steklov eigenvalue, providing a nonlocal approximation of the classical boundary value problem via fractional calculus.

ABSTRACT

In this paper we analyze possible extensions of the classical Steklov eigenvalue problem to the fractional setting. In particular, we find a nonlocal eigenvalue problem of fractional type that approximate, when taking a suitable limit, the classical Steklov eigenvalue problem.

Motivation & Objective

• To develop a nonlocal eigenvalue problem based on the fractional p-Laplacian that approximates the classical Steklov eigenvalue problem in the limit $ s \to 1^{-} $.
• To establish a connection between fractional nonlocal operators and the classical Steklov problem through asymptotic analysis of eigenvalues.
• To prove that the first eigenvalue of the fractional problem converges to the first Steklov eigenvalue as the fractional parameter approaches one.
• To rigorously analyze the convergence of the fractional eigenvalue problem to the classical Steklov problem using trace and energy estimates.

Proposed method

• Formulates a nonlocal eigenvalue problem involving the fractional p-Laplacian $ (-\Delta)^s_p $ and a boundary-weighted term $ \frac{\lambda}{\varepsilon} \chi_{\Omega_\varepsilon} |u|^{p-2}u $ in $ \Omega $, with a nonlocal Neumann-type condition $ \mathcal{N}_{s,p}u = 0 $ in $ \Omega^c $.
• Uses the normalization constant $ \mathcal{K}_{n,p} $ to ensure convergence of the fractional seminorm to the classical Dirichlet energy as $ s \to 1^{-} $.
• Employs a weak formulation based on integration by parts and the nonlocal divergence theorem to derive the variational structure of the problem.
• Introduces a normalized eigenvalue functional $ \lambda_{1,\varepsilon}(s,p) $ involving the fractional energy and a boundary strip norm over $ \Omega_\varepsilon $, with $ \varepsilon = 1-s $.
• Applies compactness and convergence results from fractional Sobolev spaces, including the convergence of fractional seminorms to the classical $ W^{1,p} $-seminorm.
• Uses trace estimates and uniform bounds on eigenfunctions to prove convergence of the eigenvalue functional to the classical Steklov eigenvalue.

Experimental results

Research questions

• RQ1 Can a fractional eigenvalue problem be constructed such that its first eigenvalue converges to the classical Steklov eigenvalue as $ s \to 1^{-} $?
• RQ2 How does the fractional p-Laplacian with a boundary-weighted term approximate the classical Steklov problem in the limit $ s \to 1 $?
• RQ3 What is the role of the normalization constant $ \mathcal{K}_{n,p} $ in bridging fractional and classical Sobolev norms?
• RQ4 Does the eigenvalue functional of the fractional problem converge to the classical Steklov eigenvalue functional under appropriate normalization?
• RQ5 What are the necessary compactness and convergence properties of eigenfunctions in the fractional setting to ensure convergence of eigenvalues?

Key findings

• The first eigenvalue $ \lambda_{1,\varepsilon}(s,p) $ of the fractional eigenvalue problem converges to the first Steklov eigenvalue $ \lambda_1(p) $ as $ s \to 1^{-} $, with $ \varepsilon = 1-s $.
• The fractional energy term $ \mathcal{K}_{n,p}(1-s)[u]_{s,p}^p $ converges to the classical Dirichlet energy $ \|\nabla u\|_{L^p(\Omega)}^p $ as $ s \to 1^{-} $.
• The boundary term $ \frac{1}{1-s}\|u\|_{L^p(\Omega_{1-s})}^p $ converges to the $ L^p $-trace norm $ \|u\|_{L^p(\partial\Omega)}^p $ as $ s \to 1^{-} $.
• The limit of the eigenvalue functional satisfies $ \limsup_{s \to 1^{-}} \lambda_{1}(s,p) \leq \lambda_1(p) $, and the reverse inequality is proven via compactness and trace convergence.
• The convergence of eigenfunctions in $ L^p(\Omega) $ and $ W^{s_0,p}(\Omega) $ for $ s_0 < 1 $ ensures the limit eigenfunction belongs to $ W^{1,p}(\Omega) $.
• The proof relies on uniform bounds on eigenfunctions and the application of the nonlocal divergence theorem and integration by parts in the fractional setting.