On the Strong Maximum Principle for Fully Nonlinear Degenerate Elliptic Equations
نویسنده
چکیده
We prove a strong maximum principle for semicontinuous viscosity subsolutions or supersolutions of fully nonlinear degenerate elliptic PDE's, which complements the results of 17]. Our assumptions and conclusions are diierent from those in 17], in particular our maximum principle implies the nonexistence of a dead core. We test the assumptions on several examples involving the p-Laplacian and the minimal surface operator, and they turn out to be sharp in all cases where the existence of a dead core is known. We can also cover equations that are singular for p = 0 and very degenerate operators such as the 1-Laplacian and some rst order Hamilton-Jacobi operators.
منابع مشابه
Gradient bounds for nonlinear degenerate parabolic equations and application to large time behavior of systems
We obtain new oscillation and gradient bounds for the viscosity solutions of fully nonlinear degenerate elliptic equations where the Hamiltonian is a sum of a sublinear and a superlinear part in the sense of Barles and Souganidis (2001). We use these bounds to study the asymptotic behavior of weakly coupled systems of fully nonlinear parabolic equations. Our results apply to some “asymmetric sy...
متن کاملComparison principles and Lipschitz regularity for some nonlinear degenerate elliptic equations
We establish interior Lipschitz regularity for continuous viscosity solutions of fully nonlinear, conformally invariant, degenerate elliptic equations. As a by-product of our method, we also prove a weak form of the strong comparison principle, which we refer to as the principle of propagation of touching points, for operators of the form ∇2ψ + L(x, ψ,∇ψ) which are non-decreasing in ψ.
متن کاملExistence and Multiplicity Results for Degenerate Elliptic Equations with Dependence on the Gradient
Recommended by Shujie Li We study the existence of positive solutions for a class of degenerate nonlinear elliptic equations with gradient dependence. For this purpose, we combine a blowup argument, the strong maximum principle, and Liouville-type theorems to obtain a priori estimates. This is an open access article distributed under the Creative Commons Attribution License, which permits unres...
متن کاملA “maximum Principle for Semicontinuous Functions” Applicable to Integro-partial Differential Equations
We formulate and prove a non-local “maximum principle for semicontinuous functions” in the setting of fully nonlinear and degenerate elliptic integro-partial differential equations with integro operators of second order. Similar results have been used implicitly by several researchers to obtain comparison/uniqueness results for integro-partial differential equations, but proofs have so far been...
متن کاملHarnack Inequalities and ABP Estimates for Nonlinear Second-Order Elliptic Equations in Unbounded Domains
In this paper we are concerned with fully nonlinear uniformy elliptic operators with a superlinear gradient term. We look for local estimates, such as weak Harnack inequality and local Maximum Principle, and their extension up to the boundary. As applications, we deduce ABP type estimates and weak Maximum Principles in general unbounded domains, a strong Maximum principle and a Liouville type t...
متن کامل