Related papers: Quantitative homogenisation for differential equat…

We construct and analyze approximation rates of deep operator networks (ONets) between infinite-dimensional spaces that emulate with an exponential rate of convergence the coefficient-to-solution map of elliptic second-order partial…

We consider a class of constant-coefficient partial differential operators on a finite-dimensional real vector space which exhibit a natural dilation invariance. Typically, these operators are anisotropic, allowing for different degrees in…

In $L_2({\mathbb R}^d;{\mathbb C}^n)$, we study a selfadjoint strongly elliptic operator $A_\varepsilon$ of order $2p$ given by the expression $b({\mathbf D})^* g({\mathbf x}/\varepsilon) b({\mathbf D})$, $\varepsilon >0$. Here $g({\mathbf…

This paper investigates quantitative estimates in elliptic homogenization of non-divergence form with unbounded drift and an interface, which continues the study of the previous work by Hairer and Manson [Ann. Probab. 39(2011) 648-682],…

This paper investigates quantitative estimates in the homogenization of second-order elliptic systems with periodic coefficients that oscillate on multiple separated scales. We establish large-scale interior and boundary Lipschitz estimates…

This paper aims at an accurate and efficient computation of effective quantities, e.g., the homogenized coefficients for approximating the solutions to partial differential equations with oscillatory coefficients. Typical multiscale methods…

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 critical sets of solutions $u_\e$ of second-order elliptic equations in divergence form with rapidly oscillating and periodic coefficients. We show that the $(d-2)$-dimensional Hausdorff measures of the critical sets…

For a class of non-selfadjoint semiclassical pseudodifferential operators with double characteristics, we study bounds for resolvents and estimates for low lying eigenvalues. Specifically, assuming that the quadratic approximations of the…

This paper is about the homogenization of linear elliptic operators in divergence form with stationary random coefficients that have only slowly decaying correlations. It deduces optimal estimates of the homogenization error from optimal…

The operator \[ A_{\varepsilon}= D_{1} g_{1}(x_{1}/\varepsilon, x_{2}) D_{1} + D_{2} g_{2}(x_{1}/\varepsilon, x_{2}) D_{2} \] is considered in $L_{2}({\mathbb{R}}^{2})$, where $g_{j}(x_{1},x_{2})$, $j=1,2,$ are periodic in $x_{1}$ with…

In this work we consider a system of quasilinear elliptic equations driven by an anisotropic $p$-Laplacian. The lower-order nonlinearities are in potential form and exhibit critical Sobolev growth. We exhibit conditions on the coefficients…

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 prove an asymptotic expansion for the eigenvalues and eigenfunctions of Schr\"{o}dinger-type operator with a confining potential and with principle part a periodic elliptic operator in divergence form. We compare the spectrum to the…

In this paper, we study the nonexpansive properties of metric resolvent, and present a convergence rate analysis for the associated fixed-point iterations (Banach-Picard and Krasnosel'skii-Mann types). Equipped with a variable metric, we…

This paper addresses the issue of homogenization of linear divergence form parabolic operators in situations where no ergodicity and no scale separation in time or space are available. Namely, we consider divergence form linear parabolic…

The convergence of an adaptive mixed finite element method for general second order linear elliptic problems defined on simply connected bounded polygonal domains is analyzed in this paper. The main difficulties in the analysis are posed by…

We obtained estimates for first eigenvalues of the Dirichlet boundary value problem for elliptic operators in divergence form (i.e. for the principal frequency of non-homogeneous membranes) in bounded domains $\Omega \subset \mathbb C$…

This paper was devoted to study the quantitative homogenization problems for nonlinear elliptic operators in perforated domains. We obtained a sharp error estimate $O(\varepsilon)$ when the problem was anchored in the reference domain…

In this study, we prove results on the weak solvability and homogenization of a microscopic semi-linear elliptic system posed in perforated media. The model presented here explores the interplay between stationary diffusion and both surface…