Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators

2013/10/11 by Henri Berestycki, Italo Capuzzo Dolcetta, Berestycki, Henri +5
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1310.3192

openalex publication_date 2013/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We characterize the validity of the Maximum Principle in bounded domains for fully nonlinear degenerate elliptic operators in terms of the sign of a suitably defined generalized principal eigenvalue. Here, maximum principle refers to the non-positivity of viscosity subsolutions of the Dirichlet problem. This characterization is derived in terms of a new notion of generalized principal eigenvalue, which is needed because of the possible degeneracy of the operator, admitted in full generality. We further discuss the relations between this notion and other natural generalizations of the classical notion of principal eigenvalue, some of which had already been used in the literature for particular classes of operators.

Related

• Maximum Principle and principal eigenvalue in unbounded domains under general boundary conditions
• Generalizations and properties of the principal eigenvalue of elliptic\n operators in unbounded domains
• Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators
• Maximum and comparison principles for degenerate elliptic systems and\n some applications
• A family of degenerate elliptic operators: maximum principle and its consequences