Harnack Inequalities and ABP Estimates for Nonlinear Second‐Order Elliptic Equations in Unbounded Domains

Abstract

We are concerned with fully nonlinear uniformly 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 theorem.

Liouville properties and critical value of fully nonlinear elliptic operators

“…This e some results in [18]. Let us mention that other results on Liouville type properties for fully nonlinear equations are in the papers [14,3,40,6,39]. The second part of the paper is devoted to two applications of the Liouville properties, both for uniformly elliptic F .…”

“…Assume G satisfies the conditions (1), (2),(3).If u ∈ U SC([0, +∞) × R N ) satisfies u t + G[u] ≤ 0 in R N × (0, +∞),lim sup |x|→+∞ u(t, x) w(x) ≤ 0 uniformly in t ∈ [0, +∞), and either c ≡ 0 or u ≥ 0, then lim sup t→+∞,y→x u(t, y) =: u(x)…”

Regularity Properties for a Class of Non-uniformly Elliptic Isaacs Operators

“…As an application of the Harnack inequality, we get at once the Liouville result of Theorem 1.7. The same technique of uniformly elliptic operators can be used, which is only based on the Harnack inequality See for instance [2].…”

Positive solution to extremal Pucci's equations with singular and gradient nonlinearity

“…In case α = 0, (1) has been of independent interest. Such type of problems first of all appear in the thesis [24] and later have been studied by many authors, see [32,33,34,44].…”