[Paper Review] The anisotropic $\infty$-Laplacian eigenvalue problem with Neumann boundary conditions

Gianpaolo Piscitelli|arXiv (Cornell University)|Jul 2, 2017

Advanced Mathematical Modeling in Engineering17 references5 citations

TL;DR

This paper investigates the limit of the anisotropic p-Laplacian eigenvalue problem with Neumann boundary conditions as p → ∞, establishing that the first nontrivial eigenvalue Λ∞(Ω) is bounded below by 2/diamF(Ω). It proves a Szegö-Weinberger-type inequality showing that the Wulff shape maximizes this eigenvalue under fixed measure, and demonstrates that the first ∞-eigenfunction attains its extrema only on the boundary, with no internal nodal lines in convex domains.

ABSTRACT

We analize the limit problem of the anisotropic $p$-Laplacian as $p ightarrow\infty$ with the mean of the viscosity solution. We also prove some geometric properties of eigenvalues and eigenfunctions. In particular, we show the validity of a Szeg\"o-Weinberger type inequality.

Motivation & Objective

• To analyze the asymptotic behavior of the anisotropic p-Laplacian eigenvalue problem with Neumann boundary conditions as p → ∞.
• To characterize the first nontrivial eigenvalue Λ∞(Ω) in the limit and establish its geometric lower bound.
• To prove a Szegö-Weinberger-type inequality for the first ∞-eigenvalue in convex domains.
• To investigate geometric properties of ∞-eigenfunctions, including the location of extrema and the absence of internal nodal lines.

Proposed method

• Uses viscosity solution theory to analyze the limiting problem of the anisotropic p-Laplacian as p → ∞.
• Defines the limiting operator Q∞u = F²(∇u)(∇²u ∇ξF(∇u)) · ∇ξF(∇u), which governs the ∞-Laplacian behavior.
• Applies the concept of Finsler metric via a convex, 1-homogeneous norm F(ξ), with dual norm Fo(ξ).
• Introduces the anisotropic inradius iF(Ω) and diameter diamF(Ω) to characterize extremal geometric properties.
• Employs approximation arguments and comparison principles to relate Neumann and Dirichlet eigenvalues.
• Utilizes symmetrization and rearrangement techniques to prove the Szegö-Weinberger inequality.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of the first nontrivial Neumann eigenvalue of the anisotropic p-Laplacian as p → ∞?
• RQ2 Does the Wulff shape maximize the first ∞-eigenvalue among sets of fixed measure, as in the Szegö-Weinberger inequality?
• RQ3 Can the first ∞-eigenfunction for the Neumann problem attain its maximum or minimum only on the boundary of a convex domain?
• RQ4 Are there any closed nodal lines in the domain for the first nontrivial ∞-eigenfunction in convex domains?
• RQ5 How does the first nontrivial Neumann ∞-eigenvalue compare to the first Dirichlet ∞-eigenvalue?

Key findings

• The first nontrivial Neumann eigenvalue Λ∞(Ω) satisfies Λ∞(Ω) ≥ 2/diamF(Ω), with equality if and only if Ω is a Wulff shape.
• The Wulff shape Ω# maximizes Λ∞(Ω) among all sets of equal measure, proving a Szegö-Weinberger-type inequality.
• The first nontrivial Neumann ∞-eigenvalue is never larger than the first Dirichlet ∞-eigenvalue: Λ∞(Ω) ≤ λ∞(Ω), with equality iff Ω is a Wulff shape.
• No closed nodal lines can exist in the domain for the first ∞-eigenfunction; the function cannot vanish on a positive-measure interior set while being non-zero elsewhere.
• The maximum and minimum of the first ∞-eigenfunction are attained only on the boundary ∂Ω, and the points achieving them are at maximal anisotropic distance.
• The extrema are located at points x and x̄ such that Fo(x − x̄) = diamF(Ω), confirming the geometric extremality of boundary points.