Related papers: A Carleman-Type Inequality in Elliptic Periodic Ho…
In this paper we prove a general homogenization result for monotone parabolic problems with an arbitrary number of microscopic scales in space as well as in time, where the scale functions are not necessarily powers of epsilon. The main…
In this paper, we establish the higher order convergence rates in periodic homogenization of viscous Hamilton-Jacobi equations, which is convex and grows quadratically in the gradient variable. We observe that although the nonlinear…
It is shown that natural boundary conditions for non-relativistic wave functions are of periodic or of homogeneous Robin type. Using asymptotic central symmetry of Hamiltonian and theory of singular differential equations the many-electron…
In this paper, we derive a boundary and an internal observability inequality for stochastic hyperbolic equations with nonsmooth lower order terms. The required inequalities are obtained by global Carleman estimate for stochastic hyperbolic…
This paper is devoted to logarithmic Hardy-Littlewood-Sobolev inequalities in the two-dimensional Euclidean space, in presence of an external potential with logarithmic growth. The coupling with the potential introduces a new parameter,…
The paper deals with homogenization and higher order approximations of solutions to nonlocal evolution equations of convolution type whose coefficients are periodic in the spatial variables and random stationary in time. We assume that the…
This paper focuses on the uniform boundary estimates in homogenization of a family of higher order elliptic operators $\mathcal{L}_\epsilon$, with rapidly oscillating periodic coefficients. We derive uniform boundary $C^{m-1,\lambda}…
In the elliptic theory for $p$-Laplacian-like problems, the H\"{o}lder continuity of solutions has been proven for problems arising as Euler--Lagrange equations of a convex potential with $p$-growth that additionally satisfies the splitting…
In this paper, we suggest a new heterogeneous multiscale method (HMM) for the time-harmonic Maxwell equations in locally periodic media. The method is constructed by using a divergence-regularization in one of the cell problems. This allows…
The purpose of this work is to investigate the exponential stability of a second order coupled wave equations by laplacian with one locally internal viscous damping. Firstly, using a unique continuation theorem combined with a Carleman…
We study optimal convergence rates in the periodic homogenization of linear elliptic equations of the form $-A(x/\varepsilon):D^2 u^{\varepsilon} = f$ subject to a homogeneous Dirichlet boundary condition. We show that the optimal rate for…
We prove the homogenization of the Dirichlet problem for fully nonlinear elliptic operators with periodic oscillation in the operator and of the boundary condition for a general class of smooth bounded domains. This extends the previous…
We consider a non-uniformly elliptic second-order differential operator with periodic coefficients that models composite media consisting of highly anisotropic cylindrical fibres periodically distributed in an isotropic background. The…
In this paper a second-order homogenization approach for periodic material is derived from an appropriate representation of the down-scaling that correlates the microdisplacement field to the macro-displacement field and the macro-strain…
This paper shows that the Heterogeneous Multiscale Method can be applied to elliptic problem without scale separation. The Localized Orthogonal Method is a special case of the Heterogeneous Multiscale Method.
This article studies the homogenization of hyperbolic-parabolic equations in porous media with tiny holes. We assume that the holes are periodically distributed and that the coefficients of the equations are periodic. Using the multi-scale…
In this paper, we analyze a family of hybridizable discontinuous Galerkin (HDG) methods for second order elliptic problems in two and three dimensions. The methods use piecewise polynomials of degree $k\geqslant 0$ for both the flux and…
We study local regularity properties for solutions of linear, non-uniformly elliptic equations. Assuming certain integrability conditions on the coefficient field, we prove local boundedness and Harnack inequality. The assumed integrability…
In this paper we investigate the homogenization problem with a non-homogeneous Dirichlet condition. Our aim is to give error estimates with boundary data in $H^{1/2}(\partial\Omega)$. The tools used are those of the unfolding method in…
This paper concerns the boundary behavior of solutions of certain fully nonlinear equations with a general drift term. We elaborate on the non-homogeneous generalized Harnack inequality proved by the second author in (Julin, ARMA -15), to…