Liouville properties and critical value of fully nonlinear elliptic operators

Abstract

Abstract. We prove some Liouville properties for sub-and supersolutions of fully nonlinear degenerate elliptic equations in the whole space. Our assumptions allow the coefficients of the first order terms to be large at infinity, provided they have an appropriate sign, as in OrnsteinUhlenbeck operators. We give two applications. The first is a stabilization property for large times of solutions to fully nonlinear parabolic equations. The second is the solvability of an ergodic Hamilton-Jacobi-Bellman equation that identifies a unique critical value of the operator.

Liouville results for fully nonlinear equations modeled on Hörmander vector fields: I. The Heisenberg group

“…Here these terms must be large for large |x|, contrary to the results quoted above. In the case of Pucci's operators the results of [5] are different from those in [20] and fit better the treatment of uniformly elliptic equations via the inequalities (4). In Sect.…”

“…A new approach to Liouville properties for sub-and supersolutions of Hamilton-Jacobi-Bellman elliptic equations involving operators of Ornstein-Uhlenbeck type was introduced in [6], based on the strong maximum principle and the existence of a sort of Lyapunov function for the equation. It was applied in [5] to fully nonlinear uniformly elliptic equations of the form (1) and to some quasilinear hypoelliptic equations, under assumptions on the sign of the coefficients of the first and zero-th order terms, and on their size. Here these terms must be large for large |x|, contrary to the results quoted above.…”

New strong maximum and comparison principles for fully nonlinear degenerate elliptic PDEs

“…Definitions and preliminaries. We begin by comparing our Definition 1.1 of subunit vector for the operator F with the classical one given by Fefferman-Phong for linear operators (2). We recall that a vector Z is subunit for A at a point x, that we freeze and do not display, if A ≥ Z ⊗ Z(x).…”

“…The name is motivated by the the notion introduced by Fefferman and Phong [18] for linear operators (2) F (x, D 2 u(x)) := −Tr(A(x)D 2 u(x)).…”

Strictly convex solutions for singular Monge–Ampère equations with nonlinear gradient terms: existence and boundary asymptotic behavior

“…Moreover, it is well known that the Monge-Ampère operator is a fully nonlinear partial differential operator, and we notice that some fully nonlinear elliptic operators have attracted the attention of Dai [29], Jiang, Trudinger and Yang [30], Guan and Jiao [31], Jian, Wang and Zhao [32], Ji and Bao [33], Caffarelli, Li and Nirenberg [34], Amendola, Galise and Vitolo [35], Galise and Vitolo [36], Capuzzo-Dolcetta, Leoni and Vitolo [37], Bardi and Cesaroni [38], and Lazer and McKenna [25]. For the other latest related papers, see Zhang [39], and Feng and Zhang [40].…”