97 research outputs found
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 , 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
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