G˚arding Cones and Bellman Equations in the Theory of Hessian Operators and Equations

In this work, we continue investigation of algebraic properties of G˚arding cones in the space of symmetric matrices. Based on this theory, we propose a new approach to study of fully nonlinear differential operators and second-order partial differential equations. We prove new-type comparison theorems for evolution Hessian operators and establish a relation between Hessian and Bellman equations.

Homogenization of monotone type problems with uncertain data

The paper deals with homogenization of nonlinear differential operators with monotone behaviour. We consider a situation, when the coefficients of the operator are not known exactly, but in certain bounds only due to errors caused by measurements. We use the deterministic approach to the problem- -worst scenario method introduced by I. Hlaváček.

On invariant subspaces for nonlinear finite-difference operators

We study linear subspaces invariant under discrete operators corresponding to finitedifference approximations of differential operators with polynomial nonlinearities. In several cases, we establish a certain structural stability of invariant subspaces and sets of nonlinear differential operators of reaction–diffusion type with respect to their spatial discretisation. The corresponding lower-dimensional reductions of the finite-difference solutions on the invariant subspaces are constructed.