Related papers: Parabolic Homogenization with an Interface

This note is a summary of the recent paper [9]. Here, we study the homogenization of elliptic systems with Dirichlet boundary condition, when both the coefficients and the boundary datum are oscillating. In particular, in the paper [9], we…

The interface problem for the linear Schr\"odinger equation in one-dimensional piecewise homogeneous domains is examined by providing an explicit solution in each domain. The location of the interfaces is known and the continuity of the…

In this paper we homogenise monotone parabolic problems with two spatial scales and finitely many temporal scales. Under a certain well-separatedness assumption on the spatial and temporal scales as explained in the paper, we show that…

We study the quantitative homogenization of linear second order elliptic equations in non-divergence form with highly oscillating periodic diffusion coefficients and with large drifts, in the so-called ``centered'' setting where…

In multi-phase fluid flow, fluid-structure interaction, and other applications, partial differential equations (PDEs) often arise with discontinuous coefficients and singular sources (e.g., Dirac delta functions). These complexities arise…

In this paper, we are interested in the reiterated homogenization of linear elliptic equations of the form $-\frac{\partial}{\partial x_{i}} \left(a_{i j} \left(\frac{x}{\varepsilon}, \frac{x}{\varepsilon^{2}}\right) \frac{\partial…

We analyze a homogenization limit for the linear wave equation of second order. The spatial operator is assumed to be of divergence form with an oscillatory coefficient matrix $a^\varepsilon$ that is periodic with characteristic length…

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…

Homogenization of a scalar elliptic equation in a bounded domain with Neuman boundary condition is studied. Coefficients of the operator are oscillating over two different groups of variables with different small periods $\varepsilon$ and…

The work is motivated by the Faraday cage effect. We consider the Helmholtz equation over a 3D-domain containing a thin heterogeneous interface of thickness $\delta \ll 1$. The layer has a $\delta-$periodic structure in the in-plane…

In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider a selfadjoint matrix strongly elliptic second order differential operator $\mathcal{A}_\varepsilon$, $\varepsilon >0$. The coefficients of the operator $\mathcal{A}_\varepsilon$ are periodic…

In the first part of this series, an augmented PDE system was introduced in order to couple two nonlinear hyperbolic equations together. This formulation allowed the authors, based on Dafermos's self-similar viscosity method, to establish…

We present an efficient method for the computation of homogenized coefficients of divergence-form operators with random coefficients. The approach is based on a multiscale representation of the homogenized coefficients. We then implement…

We consider the Dirichlet problem for elliptic systems with periodically distributed inclusions whose conduction parameter exhibits a significant contrast compared to the background media. We develop a unified method to quantify the…

We consider divergence form elliptic operators in dimension $n\geq 2$ with $L^\infty$ coefficients. Although solutions of these operators are only H\"{o}lder continuous, we show that they are differentiable ($C^{1,\alpha}$) with respect to…

In a planar infinite strip with a fast oscillating boundary we consider an elliptic operator assuming that both the period and the amplitude of the oscillations are small. On the oscillating boundary we impose Dirichlet, Neumann or Robin…

We approximate an elliptic problem with oscillatory coefficients using a problem of the same type, but with constant coefficients. We deliberately take an engineering perspective, where the information on the oscillatory coefficients in the…

We establish interior Lipschitz estimates at the macroscopic scale for solutions to systems of linear elasticity with rapidly oscillating periodic coefficients and mixed boundary conditions in domains periodically perforated at a…

This paper is devoted to studying a type of contact problems modeled by hemivariational inequalities with small periodic coefficients appearing in PDEs, and the PDEs we considered are linear, second order and uniformly elliptic. Under the…

Here, we study the periodic homogenization problem of nonlinear weakly coupled systems of Hamilton-Jacobi equations in the convex setting. We establish a rate of convergence $O(\sqrt{\varepsilon})$ which is sharp.