Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators

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, the maximum principle refers to the property of non-positivity of viscosity subsolutions of the Dirichlet problem. The new notion of generalized principal eigenvalue that we introduce here allows us to deal with arbitrary type of degeneracy of the elliptic operators. We further discuss the relations between this notion and other natural generalizations of the classical notion of principal eigenvalue, some of which have been previously introduced for particular classes of operators

Maximum Principles and Aleksandrov--Bakelman--Pucci Type Estimates for NonLocal Schrödinger Equations with Exterior Conditions

“…Indeed, v(x) corresponds to the mean of the first exit time of the diffusion generated by L − c starting from a point x ∈ D, and thus boundary conditions are set only on those points of ∂D which can be reached by an exit event through a sequence (x n ) n∈N . For subsequent developments on the characterization of the generalized principal eigenvalue and further generalizations we refer to [5,7,59,62], and for a book-length discussion of the probabilistic aspects to [61].…”

The Brunn–Minkowski inequality for the principal eigenvalue of fully nonlinear homogeneous elliptic operators

“…). -A continuous function u : Ω → R is a viscosity solution to (6) in Ω if it is both a viscosity super-solution and a viscosity sub-solution.…”

“…The bibliography related to the eigenvalue problem for fully nonlinear second order operators is very wide. With no attempt of completeness, we limit ourselves to quote Birindelli-Demengel for many related works including [11,[13][14][15], Ikoma-Ishii for the computation of eigenvalues on balls [45,46], Armstrong [2], Berestycki-Capuzzo Dolcetta-Porretta-Rossi [6], and Quaas-Sirakov [63] for related maximum principles, Berestycki-Rossi for the case of unbounded domains [8], Kawohl and different coauthors for the case of the game theoretic p-Laplacian [5,[49][50][51][52] (see also our recent joint work [37]), Juutinen for the case of the normalized infinity Laplacian [48], Busca-Esteban-Quaas for the case of Pucci operators [24]. As far as we know, there is no previous attempt to prove that the Brunn-Minkowski inequality holds true for the principal eigenvalue of a fully nonlinear operator.…”

Maximum and comparison principles to Lane‐Emden systems

“…Great developing has also been achieved in the study of maximum principles and principal eigenvalues for other important non-linear differential operators of second order such as positively homogeneous fully non-linear operators (see [7,9,10,12,16,23,26,32,36,47]) which include models as Pucci and p-Laplace-type operators, among others. Interesting vector counterparts of (1) have also been investigated during the past years, that is, eigenvalue problems for elliptic systems.…”