On the weak maximum principle for fully nonlinear elliptic pde's in general unbounded domains

The aim of this Note is to review some recent research on viscosity solutions of fully nonlinear equations of the form F x; u(x);Du(x);D2u(x) = 0 ; x 2 where is an open set in IRN and F is a nonlinear function of its entries which is elliptic with respect to the Hessian matrix D2u of the unknown function u and satises some suitable structure condition. The main issues touched here are the Alexandrov-Bakelman-Pucci estimate, the weak Maximum Principle for bounded solutions in general unbounded domains and qualitative Phragmen-Lindel of type theorems

Representations of solutions of Hamilton-Jacobi equations

In this paper we report on some classical and more recent results about representation formulas for generalized solutions of the evolution partial differential equation ut + H(x,Du) = 0 , (x, t) 2 IRN × (0,+1) (1.1) We consider here only the case where H = H(x, p) is a convex function with respect to the p variable. In this setting, representation formulas can be obtained by exploiting the well - known connection existing via convex duality between the Hamilton - Jacobi equation (1.1) with Calculus of Variations or, more generally, Optimal Control problems

On the approximation of the principal eigenvalue for a class of nonlinear elliptic operators

We present a finite difference method to compute the principal eigenvalue and the corresponding eigenfunction for a large class of second order elliptic operators including notably linear operators in nondivergence form and fully nonlinear operators. The principal eigenvalue is computed by solving a finite-dimensional nonlinear min-max optimization problem. We prove the convergence of the method and we discuss its implementation. Some examples where the exact solution is explicitly known show the effectiveness of the method

Mean field games: convergence of a finite difference method

Mean field type models describing the limiting behavior, as the number of players tends to $+\infty$, of stochastic differential game problems, have been recently introduced by J-M. Lasry and P-L. Lions. Numerical methods for the approximation of the stationary and evolutive versions of such models have been proposed by the authors in previous works . Convergence theorems for these methods are proved under various assumption

Some properties of solutions of fully nonlinear partial differential inequalities

Invited review paper for a Volume in honor of O.A. Ladyzhenskay

Entire subsolutions of fully nonlinear degenerate elliptic equations

We prove existence and non existence results for fully nonlinear degenerate elliptic inequalities, by showing that the classical Keller--Osserman condition on the zero order term is a necessary and sufficient condition for the existence of entire sub solutions

Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators

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, maximum principle refers to the non-positivity of viscosity subsolutions of the Dirichlet problem. This characterization is derived in terms of a new notion of generalized principal eigenvalue, which is needed because of the possible degeneracy of the operator, admitted in full generality. We further discuss the relations between this notion and other natural generalizations of the classical notion of principal eigenvalue, some of which had already been used in the literature for particular classes of operators