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The asymptotic behavior of the solution of an infinite set of Smoluchowski's discrete coagulation-fragmentation-diffusion equations with non-homogeneous Neumann boundary conditions, defined in a periodically perforated domain, is analyzed.…
We propose a numerical homogenization method for scalar linear partial differential equations with rough coefficients, that integrates classical coarse-scale solvers with quantum subroutines for fine-scale corrections. Inspired by the…
This paper presents a homogenization framework for elastomeric metamaterials exhibiting long-range correlated fluctuation fields. Based on full-scale numerical simulations on a class of such materials, an ansatz is proposed that allows to…
We develop a quantitative theory of stochastic homogenization in the more general framework of differential forms. Inspired by recent progress in the uniformly elliptic setting, the analysis relies on the study of certain subadditive…
Over the last few decades, phase-field equations have found increasing applicability in a wide range of mathematical-scientific fields (e.g. geometric PDEs and mean curvature flow, materials science for the study of phase transitions) but…
We introduce a topology ${\cal T}$ on the space $U$ of uniformly discrete subsets of the Euclidean space. Assume that $S$ in $U$ admits a unique autocorrelation measure. The diffraction measure of $S$ is purely atomic if and only if $S$ is…
This work is concerned with homogenization problems for elliptic equations of the type \[ \begin{cases} \mathfrak{L}_{\delta} u_{\delta} + \lambda u_{\delta} = f_{\delta} \qquad \text{in} \;\; D, \\ \qquad \quad \;\, u = 0 \qquad \,…
We investigate a class of quasilinear elliptic system involving a nonhomogeneous differential operator which is introduced by C. A. Stuart [Milan J. Math. 79 (2011), 327-341] and depends on not only $\nabla u$ but also $u$. We show that the…
In this work we study the homogenization problem for (nonlinear) eigenvalues of quasilinear elliptic operators. We prove convergence of the first and second eigenvalues and, in the case where the operator is independent of $\varepsilon$,…
In the whole space $R^d$, $d\ge 2$, we study homogenization of a divergence form elliptic fourth-order operator $A_\varepsilon$ with measurable $\varepsilon$-periodic coefficients, where $\varepsilon$ is a small parameter. For the resolvent…
In $L_2({\mathbb R}^3;{\mathbb C}^3)$, we consider a selfadjoint operator ${\mathcal L}_\varepsilon$, $\varepsilon >0$, given by the differential expression $\mu_0^{-1/2}\operatorname{curl} \eta(\mathbf{x}/\varepsilon)^{-1}…
For a strongly elliptic pseudodifferential operator $L$ of order $2a$ ($0<a<1$) with real kernel, we show an integration-by-parts formula for solutions of the homogeneous Dirichlet problem, in the model case where the operator is…
Multiscale periodic homogenization is extended to an Orlicz-Sobolev setting. It is shown by the reiteraded periodic two-scale convergence method that the sequence of minimizers of a class of highly oscillatory minimizations problems…
We prove quantitative homogenization results for harmonic functions on supercritical continuum percolation clusters--that is, Poisson point clouds with edges connecting points which are closer than some fixed distance. We show that, on…
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 relative stability of three-dimensional icosahedral quasicrystals in multi-component systems has been investigated based on a coupled-mode Swift-Hohenberg model with two-length-scales. A recently developed projection method, which…
We review a coarse-graining theory for divergence-form elliptic operators. The construction centers on a pair of coarse-grained matrices defined on spatial blocks that encode a scale-dependent notion of ellipticity, transmit precise…
In this paper, we study the stochastic homogenization for a family of integral functionals with convex and nonstandard growth integrands defined on Orlicz-Sobolev's spaces. One fundamental in this topic is to extend the classical…
In this paper we address three aspects of nonlinear computational homogenization of elastic solids by two-scale finite element methods. First, we present a nonlinear formulation of the finite element heterogeneous multiscale method FE-HMM…
In this paper, we consider the homogenization problem for generalized elliptic systems $$ \mathcal{L}_{\varepsilon}=-\operatorname{div}(A(x/\varepsilon)\nabla+V(x/\varepsilon))+B(x/\varepsilon)\nabla+c(x/\varepsilon)+\lambda I $$ with…