Comparison principles and Lipschitz regularity for some nonlinear degenerate elliptic equations

Abstract

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 ∇ 2 ψ + L(x, ψ, ∇ψ) which are non-decreasing in ψ.

Towards a Liouville Theorem for Continuous Viscosity Solutions to Fully Nonlinear Elliptic Equations in Conformal Geometry

“…Our proof of the strong comparison principle and Hopf Lemma uses ideas in Caffarelli, Li and Nirenberg [9] and an earlier work of the authors [33]. In fact we establish them for more general equations of the form…”

“…(Here we have used the definition ofψ ε in the last inequality.) An immediate consequence of (32)- (33) and the fact that ψ 2 is a super-solution of (9) is that either…”

Existence and nonexistence to exterior Dirichlet problem for Monge–Ampère equation

“…To proceed, let us recall the definition of viscosity solutions, see [13,15] in this context. We first define the upper semi-continous and lower semi-continuous functions.…”

“…Let u, v be two viscosity solutions of (3.1), then for any 0 < α < 1, αu + (1 − α)v is a viscosity subsolution of (3.1).Proof. To begin with, let us recall the definition of ǫ-upper envelope of u, see e.g [15]. u ǫ (x) := max y∈Ω {u(y) − 1 ǫ |y − x| 2 }, x ∈Ω.Thenu ǫ → u, ǫ → 0 + , in C 0 loc (Ω).…”

Strong comparison principles for some nonlinear degenerate elliptic equations

“…and Γ ξ − ǫ denote the convex envelope of ξ − ǫ := − min{ξ ǫ , 0} on B(x, δ). Then by (20) in [24] and (8), we have…”