Picard principle for linear elliptic differential operators

A Maximum Principle for Positive Elliptic Pseudo-differential Operators on Closed Riemannian Manifolds

In this note we establish the positivity of Green’s functions for a class of elliptic differential operators on closed, Riemannian manifolds

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, the maximum principle refers to the property of non-positivity of viscosity subsolutions of the Dirichlet problem. The new notion of generalized principal eigenvalue that we introduce here al...

Monomiality Principle and Eigenfunctions of Differential Operators

On Elliptic Differential Operators with Shifts

The motivating point of our research was differential operators on the noncommutative torus studied by Connes [1, 2], who in particular obtained an index formula for such operators. These operators include shifts (more precisely, in this case, irrational rotations); hence our interest in general differential equations with shifts naturally arose. Let M be a smooth closed manifold. We consider o...

A posteriori estimates of inverse operators for boundary value problems in linear elliptic partial differential equations

A posteriori estimates of inverse operators for boundary value problems in linear elliptic partial differential equations Abstract This paper presents constructive a posteriori estimates of inverse operators for boundary value problems in linear elliptic partial differential equations (PDEs) on a bounded domain. This type of estimates plays an important role in the numerical verification of the...