We study and characterize the optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form. We obtain that the optimal rate of convergence is either $O(\varepsilon)$ or $O(\varepsilon^2)$ depending on the diffusion matrix $A$, source term $f$, and boundary data $g$. Moreover, we show that the set of diffusion matrices $A$ that give optimal rate $O(\varepsilon)$ is open and dense in the set of $C^{2,α}$ periodic, symmetric, and positive definite matrices, which means that generically, the optimal rate is $O(\varepsilon)$.
We study the continuity/discontinuity of the effective boundary condition for periodic homogenization of oscillating Dirichlet data for nonlinear divergence form equations and linear systems. For linear systems we show continuity, for nonlinear equations we give an example of discontinuity.
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], where they investigated the limiting long time/large scale behavior of such a process under diffusive rescaling. We determine the effective equation and obtain the size estimates of the gradient of Green functions as well as the optimal (in general) convergence rates. The proof relies on transferring the non-divergence form into the divergence-form with the coe…
We introduce a new constructive method for establishing lower bounds on convergence rates of periodic homogenization problems associated with divergence type elliptic operators. The construction is applied in two settings. First, we show that solutions to boundary layer problems for divergence type elliptic equations set in halfspaces and with infinitely smooth data, may converge to their corresponding boundary layer tails as slow as one wish depending on the position of the hyperplane. Second, we construct a Dirichlet problem for divergence type elliptic operators set in a bounded domain, and…
We study the inverse of the divergence operator on a domain $Ω\subset R^3$ perforated by a system of tiny holes. We show that such inverse can be constructed on the Lebesgue space $L^p(Ω)$ for any $1< p < 3$, with a norm independent of perforation, provided the holes are suitably small and their mutual distance suitably large. Applications are given to problems arising in homogenization of steady compressible fluid flows.
We establish higher order convergence rates in the theory of periodic homogenization of both linear and fully nonlinear uniformly elliptic equations of non-divergence form. The rates are achieved by involving higher order correctors which fix the errors occurring both in the interior and on the boundary layer of our physical domain. The proof is based on a viscosity method and a new regularity theory which captures the stability of the correctors with respect to the shape of our limit profile.
In this note we comment on the homogenization of a random elliptic operator in divergence form $-\nabla \cdot a\nabla$, where the coefficient field $a$ is distributed according to a stationary, but not necessarily ergodic, probability measure $P$. We generalize the well-known case for $P$ stationary and ergodic by showing that the operator $-\nabla \cdot a(\frac{\cdot}{\varepsilon})\nabla$ almost surely homogenizes to a constant-coefficient, random operator $-\nabla \cdot A_h\nabla$. Furthermore, we use a disintegration formula for $P$ with respect to a family of ergodic and stationary probabi…
We analyze behavior of weak solutions to compressible fluid flows in a bounded domain in $\mathbb{R}^3$, randomly perforated by tiny balls with random size. Assuming the radii of the balls scale like $\varepsilon^α$, $α> 3$, with $\varepsilon$ denoting the average distance between the balls, the problem homogenize with the same limiting equation. Our main contribution is a construction of the Bogovski\uı operator, uniformly in $\varepsilon$, without any assumptions on the minimal distance between the balls.
We provide a simple proof of quenched stochastic homogenization for random environments with a mean zero, divergence-free drift under the assumption that the drift admits a stationary $L^d$-integrable stream matrix in $d\geq 3$ or an $L^{2+δ}$-integrable stream matrix in $d=2$. In addition, we prove that the environment almost surely satisfies a large-scale Hölder regularity estimate and first-order Liouville principle.
This paper generalizes the results obtained by the authors in \cite{dangHomogenizationNondiluteSuspension2021} concerning the homogenization of a non-dilute suspension of magnetic particles in a viscous flow. More specifically, in this paper, a restrictive assumption on the coefficients of the coupled equation, made in \cite{dangHomogenizationNondiluteSuspension2021}, that significantly narrowed the applicability of the homogenization results obtained, is relaxed and a new regularity of the solution of the fine-scale problem is proven. In particular, we obtain a global $L^{\infty}$-bound for t…
We prove that diffusion equations with a space-time stationary and ergodic, divergence-free drift homogenize in law to a deterministic stochastic partial differential equation with Stratonovich transport noise. In the absence of spatial ergodicity, the drift is only partially absorbed into the skew-symmetric part of the flux through the use of an appropriately defined stream matrix. This leaves a time-dependent, spatially-homogenous transport which, for mildly decorrelating fields, converges to a Brownian noise with deterministic covariance in the homogenization limit. The results apply to uni…
It was pointed out in [P.L. Lions, G. Papanicolaou, S. Varadhan, Homogenization of Hamilton-Jacobi equation, unpublished preprint (1987)] that, for first order Hamilton-Jacobi (HJ) equations, homogenization starting with affine initial data implies homogenization for general uniformly continuous initial data. The argument makes use of some properties of the HJ semi-group, in particular, the finite speed of propagation. The last property is lost for viscous HJ equations. In this paper we prove the above mentioned implication in both viscous and non-viscous cases. Our proof relies on a variant o…
Global Hölder estimates for linearized Monge-Ampère equations in divergence form with applications to dual semigeostrophic equations and periodic homogenization
In this paper, we consider periodic soft inclusions $T_ε$ with periodicity $ε$, where the solution, $u_ε$, satisfies semi-linear elliptic equations of non-divergence in $Ω_ε=Ω\setminus \bar{T}_ε$ with a Neumann data on $\partial T^{\mathfrak a}$ . The difficulty lies in the non-divergence structure of the operator where the standard energy method based on the divergence theorem can not be applied. The main object is developing a viscosity method to find the homogenized equation satisfied by the limit of $u_ε$, called as $u$, as $ε$ approaches to zero. We introduce the concept of a compatibilit…
We study the homogenization of elliptic systems of equations in divergence form where the coefficients are compositions of periodic functions with a random diffeomorphism with stationary gradient. This is done in the spirit of scalar stochastic homogenization by Blanc, Le Bris and P.-L. Lions. An application of the abstract result is given for Maxwell's equations in random dissipative bianisotropic media.
We prove quantitative estimates on the rate of convergence for the oscillating Dirichlet problem in periodic homogenization of divergence-form uniformly elliptic systems. The estimates are optimal in dimensions larger than three and new in every dimension. We also prove a regularity estimate on the homogenized boundary condition.
This paper is concerned with periodic homogenization of second-order elliptic systems in divergence form with oscillating Dirichlet data or Neumann data of first order. We prove that the homogenized boundary data belong to $W^{1, p}$ for any $1<p<\infty$. In particular, this implies that the boundary layer tails are Hölder continuous of order $α$ for any $α\in (0,1)$.
The paper is mainly concerned with an approximate three-ball inequality for solutions in elliptic periodic homogenization. We consider a family of second order operators $\mathcal{L}_ε$ in divergence form with rapidly oscillating and periodic coefficients. It is the first time such an approximate three-ball inequality for homogenization theory is obtained. It implies an approximate quantitative propagation of smallness. The proof relies on a representation of the solution by the Poisson kernel and the Lagrange interpolation technique. Another full propagation of smallness result is also shown.
In the whole space $R^d$, $d\ge 2$, we study homogenization of a divergence form elliptic operator $A_\varepsilon$ of order $2m\ge 4$ with measurable $\varepsilon$-periodic coefficients, where $\varepsilon$ is a small parameter. For the resolvent $(A_\varepsilon+1)^{-1}$, we construct an approximation with the remainder term of order $\varepsilon^2$ in the operator $(L^2{\to}H^m)$-norm, using the resolvent of the homogenized operator, solutions of several auxiliary periodic problems on the unit cube, and smoothing operators. The homogenized operator here differs from the one commonly employed …
We consider the corrector equation associated, in homogenization theory , to a linear second-order elliptic equation in divergence form --$\partial$i(aij$\partial$ju) = f , when the diffusion coefficient is a locally perturbed periodic coefficient. The question under study is the existence (and uniqueness) of the corrector, strictly sublinear at infinity, with gradient in L r if the local perturbation is itself L r , r < +$\infty$. The present work follows up on our works [7, 8, 9], providing an alternative, more general and versatile approach , based on an a priori estimate, for this well-pos…
For a family of second-order elliptic systems in divergence form with rapidly oscillating almost-periodic coefficients, we obtain estimates for approximate correctors in terms of a function that quantifies the almost periodicity of the coefficients. The results are used to investigate the problem of convergence rates. We also establish uniform Hölder estimates for the Dirichlet problem in a bounded $C^{1, α}$ domain.
In the whole space $R^d$ ($d\ge 2$), we study homogenization of a divergence-form matrix elliptic operator $L_\varepsilon$ of an arbitrary even order larger than 2 with measurable $\varepsilon$-periodic coefficients, where $\varepsilon$ is a small parameter. We constuct an approximation for the resolvent of $L_\varepsilon$ with the remainder term of order $\varepsilon^2$ in the operator $L^2$-norm. We impose no regularity conditions on the operator beyond ellipticity and boundedness of coefficients. We use two scale expansions with correctors regularized by the Steklov smoothing.
This paper is concerned with uniform measure estimates for nodal sets of solutions in elliptic homogenization. We consider a family of second-order elliptic operators $\{ \mathcal{L}_\e\}$ in divergence form with rapidly oscillating and periodic coefficients. We show that the $(d-1)$-dimensional Hausdorff measures of the nodal sets of solutions to $\mathcal{L}_\e (u_\e)=0$ in a ball in $\R^d$ are bounded uniformly in $\e>0$. The proof relies on a uniform doubling condition and approximation of $u_\e$ by solutions of the homogenized equation.
In a recent work, Bourgain gave a fine description of the expectation of solutions of discrete linear elliptic equations on $\mathbb Z^d$ with random coefficients in a perturbative regime using tools from harmonic analysis. This result is surprising for it goes beyond the expected accuracy suggested by recent results in quantitative stochastic homogenization. In this short article we reformulate Bourgain's result in a form that highlights its interest to the state-of-the-art in homogenization (and especially the theory of fluctuations), and we state several related conjectures.
Second-order two-scale expansions, a unified proof for the regularity of the correctors based on the translation invariant and a lemma for extracting $O(ε)$ from the remainder term are presented for the second order nonlinear elliptic equation with rapidly oscillating coefficients. If the data are smooth enough, the error of the zero-order (or energy) in $L^\infty$, first-order in the Hölder norm, (linear periodic case)second-order's(even first-order's) gradient (or flux) in the maximum norm,are locally $O(ε)$. It can be used in the parabolic equation.
We continue the program initiated in a previous work, of applying integro-differential methods to Neumann Homogenization problems. We target the case of linear periodic equations with a singular drift, which includes (with some regularity assumptions) divergence equations with \emph{non-co-normal} oscillatory Neumann conditions. Our analysis focuses on an induced integro-differential homogenization problem on the boundary of the domain. Also, we use homogenization results for regular Dirichlet problems to build barriers for the oscillatory Neumann problem with the singular gradient term. We no…
We use the theory of selfdual Lagrangians to give a variational approach to the homogenization of equations in divergence form, that are driven by a periodic family of maximal monotone vector fields. The approach has the advantage of using $Γ$-convergence methods for corresponding functionals just as in the classical case of convex potentials, as opposed to the graph convergence methods used in the absence of potentials. A new variational formulation for the homogenized equation is also given.
We consider discrete non-divergence form difference operators in a random environment and the corresponding process--the random walk in a balanced random environment in $\mathbb{Z}^d$ with a finite range of dependence. We first quantify the ergodicity of the environment from the point of view of the particle. As a consequence, we quantify the quenched central limit theorem of the random walk with an algebraic rate. Furthermore, we prove an algebraic rate of convergence for the homogenization of the Dirichlet problems for both elliptic and parabolic non-divergence form difference operators.
We follow-up on our works devoted to homogenization theory for linear second-order elliptic equations with coefficients that are perturbations of periodic coefficients. We have first considered equations in divergence form in [6, 7, 8]. We have next shown, in our recent work [9], using a slightly different strategy of proof than in our earlier works, that we may also address the equation --aij$\partial$iju = f. The present work is devoted to advection-diffusion equations: --aij$\partial$iju + bj$\partial$ju = f. We prove, under suitable assumptions on the coefficients aij, bj, 1 $\le$ i, j $\l…
This paper is concerned with a family of second-order elliptic systems in divergence form with rapidly oscillating periodic coefficients. We initiate the study of homogenization and boundary layers for Neumann problems with first-order oscillating boundary data. We identify the homogenized system and establish the sharp rate of convergence in $L^2$ in dimension three or higher. Sharp regularity estimates are also obtained for the homogenized boundary data in both Dirichlet and Neumann problems. The results are used to obtain a higher-order convergence rate for Neumann problems with non-oscilla…
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 $(A_\varepsilon+1)^{-1}$, acting as an operator from $L^2$ to $H^2$, we find an approximation with remainder term of order $O(\varepsilon^2)$ as $\varepsilon$ tends to $0$. Relying on this result, we construct the resolvent approximation with remainder of order $O(\varepsilon^3)$ in the operator $L^2$-norm. We employ two-scale expansions that involve smooth…
The aim of this paper is twofold. The first is to study the asymptotics of a parabolically scaled, continuous and space-time stationary in time version of the well-known Funaki-Spohn model in Statistical Physics. After a change of unknowns requiring the existence of a space-time stationary eternal solution of a stochastically perturbed heat equation, the problem transforms to the qualitative homogenization of a uniformly elliptic, space-time stationary, divergence form, nonlinear partial differential equation, the study of which is the second aim of the paper. An important step is the construc…
In this note we study periodic homogenization of Dirichlet problem for divergence type elliptic systems when both the coefficients and the boundary data are oscillating. One of the key difficulties here is the determination of the fixed boundary data corresponding to the limiting (homogenized) problem. This issue has been addressed in recent papers by D. Gérard-Varet and N. Masmoudi, and by C. Prange, however, not much is known about the regularity of this fixed data. The main objective of this note is to initiate a study of this problem, and to prove several regularity results in this connect…
In the homogenization of divergence-form equations with random coefficients, a central role is played by the corrector. We focus on a discrete space setting and on dimension 3 and more. Completing the argument started in previous work, we identify the scaling limit of the corrector, which is akin to a Gaussian free field.
We develop a new real-variable method for weighted $L^p$ estimates. The method is applied to the study of weighted $W^{1, 2}$ estimates in Lipschitz domains for weak solutions of second-order elliptic systems in divergence form with bounded measurable coefficients. It produces a necessary and sufficient condition, which depends on the weight function, for the weighted $W^{1,2}$ estimate to hold in a fixed Lipschitz domain with a given weight. Using this condition, for elliptic systems in Lipschitz domains with rapidly oscillating, periodic and VMO coefficients, we reduce the problem of weighte…
We prove a quantitative rate of homogenization for the G equation in a random setting with finite range of dependence and nonzero divergence, with explicit dependence of the constants on the Lipschitz norm of the environment. Inspired by work of Burago, Ivanov, and Novikov, the proof uses explicit bounds on the waiting time for the associated metric problem.
We prove quantitative estimates for the stochastic homogenization of linear uniformly elliptic equations in nondivergence form. Under strong independence assumptions on the coefficients, we obtain optimal estimates on the subquadratic growth of the correctors with stretched exponential-type bounds in probability. Like the theory of Gloria and Otto \cite{GO1,GO2} for divergence form equations, the arguments rely on nonlinear concentration inequalities combined with certain estimates on the Green's functions and derivative bounds on the correctors. We obtain these analytic estimates by developin…
In this paper, we extend the uniform regularity estimates obtained by M. Avellanda and F. Lin in the paper of Compactness methods in the theory of homogenization (Comm. Pure Appl. Math. 40(1987), no.6, 803-847) to the more general second order elliptic systems in divergences form, with rapidly oscillating periodic coefficients. We establish not only sharp $W^{1,p}$ estimates, Holder estimates, Lipschitz estimates and non-tangential maximal function estimates for the Dirichelt problem on a $C^{1,η}$ domain, but also a sharp $O(\varepsilon)$ convergence rate in $H_0^1(Ω)$ by virtue of the Dirich…
This note addresses the homogenization error for linear elliptic equations in divergence-form with random stationary coefficients. The homogenization error is measured by comparing the quenched Green's function to the Green's function belonging to the homogenized coefficients, more precisely, by the (relative) spatial decay rate of the difference of their second mixed derivatives. The contribution of this note is purely deterministic: It uses the expanded notion of corrector, namely the couple of scalar and vector potentials $(ϕ,σ)$, and shows that the rate of sublinear growth of $(ϕ,σ)$ at th…
The paper deals with homogenization of divergence form second order parabolic operators whose coefficients are periodic in spatial variables and random stationary in time. Under proper mixing assumptions, we study the limit behaviour of the normalized difference between solutions of the original and the homogenized problems. The asymptotic behaviour of this difference depends crucially on the ratio between spatial and temporal scaling factors. Here we study the case of self-similar parabolic diffusion scaling.
We are interested in the shape of the homogenized operator $\overline F(Q)$ for PDEs which have the structure of a nonlinear Pucci operator. A typical operator is $H^{a_1,a_2}(Q,x) = a_1(x) λ_{\min}(Q) + a_2(x)λ_{\max}(Q)$. Linearization of the operator leads to a non-divergence form homogenization problem, which can be solved by averaging against the invariant measure. We estimate the error obtained by linearization based on semi-concavity estimates on the nonlinear operator. These estimates show that away from high curvature regions, the linearization can be accurate. Numerical results show …
We derive optimal estimates in stochastic homogenization of linear elliptic equations in divergence form in dimensions $d\ge 2$. In previous works we studied the model problem of a discrete elliptic equation on $\mathbb{Z}^d$. Under the assumption that a spectral gap estimate holds in probability, we proved that there exists a stationary corrector field in dimensions $d>2$ and that the energy density of that corrector behaves as if it had finite range of correlation in terms of the variance of spatial averages - the latter decays at the rate of the central limit theorem. In this article we ext…
We show that certain linear elliptic equations (and systems) in divergence form with almost periodic coefficients have bounded, almost periodic correctors. This is proved under a new condition we introduce which quantifies the almost periodic assumption and includes (but is not restricted to) the class of smooth, quasiperiodic coefficient fields which satisfy a Diophantine-type condition previously considered by Kozlov. The proof is based on a quantitative ergodic theorem for almost periodic functions combined with the new regularity theory recently introduced by the first author and Shen for …
In recent years considerable advances have been made in quantitative homogenization of partial differential equations in the periodic and non-periodic settings. This monograph surveys the theory of quantitative homogenization for second-order linear elliptic systems in divergence form with rapidly oscillating periodic coefficients, $$ \mathcal{L}_\e =-\text{\rm div} \big (A(x/\e)\nabla \big), $$ in a bounded domain $Ω$ in $\br^d$. It begins with a review of the classical qualitative homogenization theory, and addresses the problem of convergence rates of solutions. The main body of the monogra…
We establish uniform Lipschitz estimates for second-order elliptic systems in divergence form with rapidly oscillating, almost-periodic coefficients. We give interior estimates as well as estimates up to the boundary in bounded $C^{1,α}$ domains with either Dirichlet or Neumann data. The main results extend those in the periodic setting due to Avellaneda and Lin for interior and Dirichlet boundary estimates and later Kenig, Lin, and Shen for the Neumann boundary conditions. In contrast to these papers, our arguments are constructive (and thus the constants are in principle computable) and the …
We consider homogenization problems for linear elliptic equations in divergence form. The coecients are assumed to be a local perturbation of some periodic background. We prove $W^{1,p}$ and Lipschitz convergence of the two-scale expansion, with explicit rates. For this purpose, we use a corrector adapted to this particular setting, and dened in [10, 11], and apply the same strategy of proof as Avellaneda and Lin in [1]. We also propose an abstract setting generalizing our particular assumptions for which the same estimates hold.
We study the homogenization limit of solutions to the G-equation with random drift. This Hamilton-Jacobi equation is a model for flame propagation in a turbulent fluid in the regime of thin flames. For a fluid velocity field that is statistically stationary and ergodic, we prove sufficient conditions for homogenization to hold with probability one. These conditions are expressed in terms of travel times for the associated control problem. When the spatial dimension is equal to two and the fluid velocity is divergence-free, we verify that these conditions hold under suitable assumptions about t…
A two-dimensional structural local-defect theory for scalar non-divergence advection-diffusion homogenization
SKLMS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
We study the limit as $\e\to 0$ of the entropy solutions of the equation $\p_t \ue + \dv_x[A(\frac{x}{\e},\ue)] =0$. We prove that the sequence $\ue$ two-scale converges towards a function $u(t,x,y)$, and $u$ is the unique solution of a limit evolution problem. The remarkable point is that the limit problem is not a scalar conservation law, but rather a kinetic equation in which the macroscopic and microscopic variables are mixed. We also prove a strong convergence result in $L^1_{\text{loc}}$.
Divergence-form operators with random coefficients homogenize over large scales. Over the last decade, an intensive research effort focused on turning this asymptotic statement into quantitative estimates. The goal of this note is to review one approach for doing so based on the idea of renormalization. The discussion is highly informal, with pointers to mathematically precise statements.
We consider the so-called G-equation, a level set Hamilton-Jacobi equation, used as a sharp interface model for flame propagation, perturbed by an oscillatory advection in a spatio-temporal periodic environment. Assuming that the advection has suitably small spatial divergence, we prove that, as the size of the oscillations diminishes, the solutions homogenize (average out) and converge to the solution of an effective anisotropic first-order (spatio-temporal homogeneous) level set equation. Moreover we obtain a rate of convergence and show that, under certain conditions, the averaging enhances…
We consider Cauchy problem for a divergence form second order parabolic operator with rapidly oscillating coefficients that are periodic in spatial variables and random stationary ergodic in time. As was proved in [24] and [12] in this case the homogenized operator is deterministic. The paper focuses on non-diffusive scaling, when the oscillation in spatial variables is faster than that in temporal variable. Our goal is to study the asymptotic behaviour of the normalized difference between solutions of the original and the homogenized problems.
