[Paper Review] Regularity and uniqueness of the first eigenfunction for singular fully non linear operators

Isabeau Birindelli, Françoise Demengel|ArXiv.org|Sep 21, 2009

Advanced Mathematical Physics Problems13 references4 citations

TL;DR

This paper establishes the regularity and uniqueness of the first eigenfunction for a class of singular fully nonlinear operators, proving that the principal eigenvalue is simple and isolated under general domain conditions. Using viscosity solutions, Hölder regularity, and a strong comparison principle, the authors show that any positive eigenfunction is a positive multiple of the first eigenfunction, extending known results beyond variational settings to non-variational, fully nonlinear operators with homogeneity like the p-Laplacian.

ABSTRACT

In this article we prove that solutions of singular fully nonlinear partial differential equations are $C^{1,β}$. We also prove the simplicity of the principal eigenvalues for the Dirichlet Problem associated to these operators using that regularity, a strict comparison principle and Sard's theorem.

Motivation & Objective

• To resolve open questions on the simplicity and isolation of principal eigenvalues for singular fully nonlinear operators.
• To extend the theory of eigenvalues beyond variational settings (e.g., p-Laplacian) to non-variational, fully nonlinear operators with homogeneity properties.
• To establish a strong comparison principle for viscosity solutions under gradient non-vanishing conditions, enabling uniqueness of the first eigenfunction.
• To prove that the first eigenfunction is unique up to scaling, even in domains with multiple boundary components, under suitable regularity and dimension constraints.

Proposed method

• Uses viscosity solution theory to define and analyze the principal eigenvalue problem for operators of the form $ F[u] = |\nabla u|^\alpha \mathcal{M}_{a,A}^+(D^2u) + h(x) \cdot \nabla u |\nabla u|^\alpha $.
• Applies a fixed-point argument to establish $ C^{1,\beta} $ regularity of solutions, which is essential for applying the Hopf lemma and gradient estimates.
• Employs the Alexandrov-Bakelman-Pucci (ABP) inequality adapted to singular operators, though notes limitations due to non-sublinearity of differences of sub- and super-solutions.
• Utilizes Sard's theorem in dimension $ N=2 $ to handle eigenfunctions with sign changes in multiply connected domains.
• Applies a strong comparison principle: if two solutions touch and one has non-vanishing gradient in a region, they must coincide.
• Uses the maximum principle and comparison with eigenfunctions on subdomains to derive contradictions when eigenfunctions change sign or eigenvalues are not extremal.

Experimental results

Research questions

• RQ1 Is the principal eigenvalue simple, i.e., does any positive eigenfunction differ from the first eigenfunction only by a positive scalar multiple?
• RQ2 Can the uniqueness of the first eigenfunction be established in domains with multiple boundary components?
• RQ3 Does the strong comparison principle hold for viscosity solutions of singular fully nonlinear equations when the gradient is bounded away from zero?
• RQ4 What conditions ensure the isolation of the principal eigenvalue in non-variational, fully nonlinear settings?
• RQ5 How does the eigenvalue structure change when the domain has multiple connected components and the eigenfunction changes sign?

Key findings

• The first eigenfunction is unique up to positive scaling in any domain with a single connected boundary component.
• In two-dimensional domains with multiple boundary components, uniqueness of the first eigenfunction holds when $ \lambda^+ \neq \lambda^- $, under the assumption that the smaller eigenvalue corresponds to a positive eigenfunction.
• For $ N=2 $, if $ \lambda^+ \neq \lambda^- $, then every eigenfunction corresponding to $ \lambda_1 = \max(\lambda^+, \lambda^-) $ is either positive or negative, implying no sign-changing eigenfunctions exist for the principal eigenvalue.
• The principal eigenvalue is isolated: any sequence of eigenvalues approaching $ \lambda_1 $ must eventually have eigenfunctions of constant sign, contradicting the existence of sign-changing eigenfunctions near $ \lambda_1 $.
• The $ C^{1,\beta} $ regularity of eigenfunctions is established via a fixed-point argument, which enables the use of the Hopf lemma and gradient estimates to prove the strong comparison principle.
• In multiply connected domains, the assumption that $ \partial\Omega^+ \cap \partial\Omega = \partial\Omega_1 $ and $ \partial\Omega^- \cap \partial\Omega = \partial\Omega_2 $ leads to a contradiction via Sard’s theorem when eigenfunctions are assumed to change sign, proving uniqueness.