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In this work, we study the effective behavior of a two-dimensional variational model within finite crystal plasticity for high-contrast bilayered composites. Precisely, we consider materials arranged into periodically alternating thin…

In this paper, we consider a two-dimensional operator with an antisymmetric integral kernel, recently introduced by Z. Avetisyan and A. Karapetyants in connection to the study of general homogeneous operators. This is the unique…

In this paper, we establish stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms on metric measure spaces under general volume doubling condition. We obtain their stable equivalent characterizations in terms…

We give in this paper a convergence result concerning parallel asynchronous algorithm with bounded delays to solve a nonlinear fixed point problems. This result is applied to calculate the solution of a strongly monotone operator. Special…

We consider a singularly perturbed second order elliptic system in the whole space. The coefficients of the systems fast oscillate and depend both of slow and fast variables. We obtain the homogenized operator and in the uniform norm sense…

In this paper we continue the study initiated in [FGN] concerning the obstacle problem for a class of parabolic non-divergence operators structured on a set of vector fields X = {X_1,...,X_q} in R^n with C^1-coefficients satisfying…

This work aims to study the rates in the context of periodic homogenization of parabolic problems with large lower order terms (both drift and potential). We demonstrate that the solution is a product of three terms: (i) a function of time,…

We consider an evolutionary problem with rapidly oscillating coefficients. This causes the problem to change frequently between a parabolic and an hyperbolic state. We prove convergence of the homogenisation process in the unit square and…

Operator-type estimates of homogenization are obtained for elliptic operators of arbitrary even order equal or greater than two. Operators under consideration are non-selfadjoint with lower-order terms.

We survey the recent development of the DeGiorgi-Nash-Moser-Aronson type theory for a class of symmetric jump processes(or equivalently, a class of symmetric integro-differential operators). We focus on the sharp two-sided estimates for the…

We introduce and study in a general setting the concept of homogeneity of an operator and, in particular, the notion of homogeneity of an integral operator. In the latter case, homogeneous kernels of such operators are also studied. The…

We consider symmetric processes of pure jump type. We prove local estimates on the probability of exiting balls, the H\"older continuity of harmonic functions and of heat kernels, and convergence of a sequence of such processes.

For a class of linear elliptic equations of general type with rapidly oscillating coefficients, we use the sigma-convergence method to prove the homogenization result and a corrector-type result. In the case of asymptotic periodic…

In this paper we establish optimal regularity estimates and smoothness of free boundaries for nonlocal obstacle problems governed by a very general class of integro-differential operators with possibly singular kernels. More precisely, in…

We present an operator-asymptotic approach to the problem of homogenization of periodic composite media in the setting of three-dimensional linearized elasticity. This is based on a uniform approximation with respect to the inverse…

The paper studies homogenization problem for a non-autonomous parabolic equation with a large random rapidly oscillating potential in the case of one dimensional spatial variable. We show that if the potential is a statistically homogeneous…

In this paper we establish periodic homogenization for Hamilton-Jacobi-Bellman (HJB) equations, associated to nonlocal operators of integro-differential type. We consider the case when the fractional diffusion has the same order as the…

A systematic and unified approach to transformations and symmetries of general second order linear parabolic partial differential equations is presented. Equivalence group is used to derive the Appell type transformations, specifically…

This is the second part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. We consider a substitute to the notion of pointwise bounds for kernels of operators which usually is a…

In this paper we consider the homogenization of the evolution problem associated with a jump process that involves three different smooth kernels that govern the jumps to/from different parts of the domain. We assume that the spacial domain…