We establish higher order convergence rates in periodic homogenization of fully nonlinear uniformly parabolic Cauchy problems accompanied with rapidly oscillating initial data. Such result is new even for linear problems. Here we construct higher order initial layer and interior correctors, which describe the oscillatory behavior near the initial and interior time zone of the domain. To construct higher order correctors, we develop a regularity theory in macroscopic scales, and prove an exponential decay estimate for initial layer correctors.
The higher order expansion requires an iteration process: successively correcting the initial layer, then the interior. This leads to a more complicated asymptotic expansion, as compared to the non-oscillating data case, and this complexity is present even in the linear case.
A notable observation for fully nonlinear operators is that even if the given operator is space-time periodic, the interior correctors become aperiodic in the time variable as we proceed with the iteration process. Moreover, each interior corrector of higher order is paired with a space-time periodic version, and the difference between the two decays exponentially fast with time.
We study the averaging behavior of nonlinear uniformly elliptic partial differential equations with random Dirichlet or Neumann boundary data oscillating on a small scale. Under conditions on the operator, the data and the random media leading to concentration of measure, we prove an almost sure and local uniform homogenization result with a rate of convergence in probability.
In this article we investigate averaging properties of fully nonlinear PDEs in bounded domains with oscillatory Neumann boundary data. The oscillation is periodic and is present both in the operator and in the Neumann data. Our main result states that, when the domain does not have flat boundary parts and when the homogenized operator is rotation invariant, the solutions uniformly converge to the homogenized solution solving a Neumann boundary problem. Furthermore we show that the homogenized Neumann data is continuous with respect to the normal direction of the boundary.
In this paper we analyze the behavior of the Laplace operator with Neumann boundary conditions in a thin domain of the type $R^ε= \{(x,y) \in \R^2; x \in (0,1), 0 < y < εG(x, x/ε)\} $ where the function G(x,y) is periodic in y of period L. Observe that the upper boundary of the thin domain presents a highly oscillatory behavior and, moreover, the height of the thin domain, the amplitude and period of the oscillations are all of the same order, given by the small parameter $ε$.
In this paper, we consider a family of second-order elliptic systems subject to a periodically oscillating Robin boundary condition. We establish the qualitative homogenization theorem on any Lipschitz domains satisfying a non-resonance condition. We also use the quantitative estimates of oscillatory integrals to obtain the dimension-dependent convergence rates in $L^2$, assuming that the domain is smooth and strictly convex.
We are interested in the averaged behavior of interfaces moving in stationary ergodic environments, with oscillatory normal velocity which changes sign. This problem can be reformulated, using level sets, as the homogenization of a Hamilton-Jacobi equation with a positively homogeneous non-coercive Hamiltonian. The periodic setting was earlier studied by Cardaliaguet, Lions and Souganidis (2009). Here we concentrate in the random media and show that the solutions of the oscillatory Hamilton-Jacobi equation converge in $L^\infty$-weak $*$ to a linear combination of the initial datum and the sol…
We establish a rate of convergence of the two scale expansion (in the sense of homogenization theory) of the solution to a highly oscillatory elliptic partial differential equation with random coefficients that are a perturbation of periodic coefficients.
We study the averaging of fronts moving with positive oscillatory normal velocity, which is periodic in space and stationary ergodic in time. The problem can be reformulated as the homogenization of coercive level set Hamilton-Jacobi equations with spatio-temporal oscillations. To overcome the difficulties due to the oscillations in time and the sublinear growth of the Hamiltonian, we first study the long time averaged behavior of the associated reachable sets using geometric arguments. The results are new for higher than one dimensions even in the space-time periodic setting.
We consider an homogenization problem for the second order elliptic equation $- Δu^{\varepsilon} + \dfrac{1}{\varepsilon} V(./\varepsilon) u^{\varepsilon} + νu^{\varepsilon} =f$ when the highly oscillatory potential $V$ belongs to a particular class of non-periodic potentials. We show the existence of an adapted corrector and prove the convergence of $u^{\varepsilon}$ to its homogenized limit.
This paper is concerned with homogenization of systems of linear elasticity with rapidly oscillating periodic coefficients. We establish sharp convergence rates in $L^2$ for the mixed boundary value problems with bounded measurable coefficients.
In this work we study the homogenization problem for nonlinear eigenvalues of quasilinear elliptic operators. We obtain an explicit order of convergence in $k$ and in $\varepsilon$ for the (variational) eigenvalues.
A multi-scale characterization of the field concentrations inside composite and polycrystalline media is developed. The analysis focuses on gradient fields associated with the intensive quantities given by the temperature and the electric potential. In the linear regime these quantities are modeled by the solution of a second order elliptic partial differential equation with oscillatory coefficients. Field concentrations are measured using the $L^p$ norm of the gradient of the solution for p>2. The analysis focuses on the case when the length scale of the heterogeneities are small relative to …
This paper studies the convergence rates in $L^2$ and $H^1$ of Dirichelt problems for Stokes systems with rapidly oscillating periodic coefficients, without any regularity assumptions on the coefficients.
This paper is concerned with the optimal convergence rate in homogenization of higher order parabolic systems with bounded measurable, rapidly oscillating periodic coefficients. The sharp $O(\va)$ convergence rate in the space $L^2(0,T; H^{m-1}(\Om))$ is obtained for both the initial-Dirichlet problem and the initial-Neumann problem. The duality argument inspired by \cite{suslinaD2013} is used here.
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…
For a family of second-order parabolic systems with bounded measurable, rapidly oscillating and time-dependent periodic coefficients, we investigate the sharp convergence rates of weak solutions in $L^2$. Both initial-Dirichlet and initial-Neumann problems are studied.
In this paper, we study the rate of convergence in periodic homogenization of scalar ordinary differential equations. We provide a quantitative error estimate between the solutions of a first-order ordinary differential equation with rapidly oscillating coefficients and the limiting homogenized solution. As an application of our result, we obtain an error estimate for the solution of some particular linear transport equations.
In this paper we provide a rate of convergence for periodic homogenization of Hamilton-Jacobi-Bellman equations with nonlocal diffusion. The result is based on the regularity of the associated effective problem, where the convexity plays a crucial role. Such regularity estimates are possible from the available literature once we provide a representation formula for the effective Hamiltonian, a result that has an independent interest.
Here, we consider periodic homogenization for time-fractional Hamilton--Jacobi equations. By using the perturbed test function method, we establish the convergence, and give estimates on a rate of convergence. A main difficulty is the incompatibility between the function used in the doubling variable method, and the non-locality of the Caputo derivative. Our approach is to provide a lemma to prove the rate of convergence without the doubling variable method with respect to the time variable, which is a key ingredient.
We study the rate of convergence in periodic homogenization for convex Hamilton--Jacobi equations with multiscales, where the Hamiltonian $H=H(x, y, p): \mathbb{R}^n \times \mathbb{T}^n \times \mathbb{R}^n \to \mathbb{R }$ depends on both of the spatial variable and the oscillatory variable. In particular, we show that for the Cauchy problem, the rate of convergence is $O(\sqrtε)$ by optimal control formulas, scale separations and curve cutting techniques. We also show the rate $O(\sqrtε)$ of homogenization for the static problem based on the same idea. Additionally, we provide examples that i…
In this work we study the convergence of an homogenization problem for half-eigenvalues and Fučík eigencurves. We provide quantitative bounds on the rate of convergence of the curves for periodic homogenization problems.
In this paper we investigate homogenization results for the principal eigenvalue problem associated to $1$-homogeneous, uniformly elliptic, second-order operators. Under rather general assumptions, we prove that the principal eigenpair associated to an oscillatory operator converges to the eigenpair associated to the effective one. This includes the case of fully nonlinear operators. Rates of convergence for the eigenvalues are provided for linear and nonlinear problems, under extra regularity/convexity assumptions. Finally, a linear rate of convergence (in terms of the oscillation parameter) …
For a homogenization problem associated to a linear elliptic operator, we prove the existence of a distributional corrector and we find an approximation scheme for the homogenized coefficients. We also study the convergence rates in the asymptotic almost periodic setting, and we show that the rates of convergence for the zero order approximation, are near optimal. The results obtained constitute a step towards the numerical implementation of results from the deterministic homogenization theory beyond the periodic setting. To illustrate this, numerical simulations based on finite volume method …
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 structure governs the first order approximation, the nonlinear effect is absorbed as an external source term of a linear equation in the second and higher order approximation. Moreover, we find that the geometric shape of the initial data has to be chosen carefully according to the effective Hamiltonian, in order to achieve the higher order convergence rates.
We develop an analytical tool which is adept for detecting shapes of oscillatory functions, is useful in decomposing homogenization problems into limit-problems for kinetic equations, and provides an efficient framework for the validation of multi-scale asymptotic expansions. We apply it first to a hyperbolic homogenization problem and transform it to a hyperbolic limit problem for a kinetic equation. We establish conditions determining an effective equation and counterexamples for the case that such conditions fail. Second, when the kinetic decomposition is applied to the problem of enhanced …
In this work we study the homogenization for eigenvalues of the fractional $p-$Laplace in a bounded domain both with Dirichlet and Neumann conditions. We obtain the convergence of eigenvalues and the explicit order of the convergence rates.
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…
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 this paper we study the quantitative homogenization of second-order parabolic systems with locally periodic (in both space and time) coefficients. The $O(\varepsilon)$ scale-invariant error estimate in $L^2(0, T; L^{\frac{2d}{d-1}}(Ω))$ is established in $C^{1, 1}$ cylinders under minimum smoothness conditions on the coefficients. This process relies on critical estimates of smoothing operators. We also develop a new construction of flux correctors in the parabolic manner and a sharp estimate for temporal boundary layers.
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 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.
We analyze the $Γ$-convergence of sequences of free-discontinuity functionals arising in the modeling of linear elastic solids with surface discontinuities, including phenomena as fracture, damage, or material voids. We prove compactness with respect to $Γ$-convergence and represent the $Γ$-limit in an integral form defined on the space of generalized special functions of bounded deformation ($GSBD^p$). We identify the integrands in terms of asymptotic cell formulas and prove a non-interaction property between bulk and surface contributions. Eventually, we investigate sequences of correspondin…
This paper is concerned with the quantitative homogenization of $2m$-order elliptic systems with bounded measurable, rapidly oscillating periodic coefficients. We establish the sharp $O(\varepsilon)$ convergence rate in $W^{m-1, p_0}$ with $p_0=\frac{2d}{d-1}$ in a bounded Lipschitz domain in $\mathbb{R}^d$ as well as the uniform large-scale interior $C^{m-1, 1}$ estimate. With additional smoothness assumptions, the uniform interior $C^{m-1, 1}$, $W^{m,p}$ and $C^{m-1, α}$ estimates are also obtained. As applications of the regularity estimates, we establish asymptotic expansions for fundament…
We carry out a comprehensive study of quantitative homogenization of second-order elliptic systems with bounded measurable coefficients that are almost-periodic in the sense of H. Weyl. We obtain uniform local $L^2$ estimates for the approximate correctors in terms of a function that quantifies the almost-periodicity of the coefficient matrix. We give a condition that implies the existence of (true) correctors. These estimates as well as similar estimates for the dual approximate correctors yield optimal or near optimal convergence rates in $H^1$ and $L^2$.The $L^2$-based Hölder and Lipschitz …
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 the convergence of $u^{\varepsilon}$ to the solution of the corresponding homogenized problem in the $W^{1,p}$-norm is $\mathcal{O}(\varepsilon)$. We further obtain optimal gradient and Hessian bounds with correction terms taken into account in the $L^p$-norm. We then provide an explicit $c$-bad diffusion matrix and use it to perform various numerical experi…
In this work, we study the aspect of Bloch wave homogenization of the new notion of convergence of microstructures represented by matrices $B^ε$ related to the classical $H$-convergence of $A^ε$ introduced in \cite{TG-MV-1}. The new macro quantity $B^{#}$ appears to incorporate the interaction between the two microstructures $A^ε,B^ε$. Here we present its Bloch spectral representation along with the homogenization result.
In this paper we construct, by means of a variational formulation, the solutions of a problem of elastodynamics which includes the effect of damage for the elastic material. The result is a wave equation with time dependent operators which represents the elastic coefficients of the material undergoing damage. The dynamics that we construct also satisfies a threshold condition with the same threshold value that characterizes the quasi-static evolution of damage (see \cite{GL}).
We study the limiting probability distribution of the homogenization error for second order elliptic equations in divergence form with highly oscillatory periodic conductivity coefficients and highly oscillatory stochastic potential. The effective conductivity coefficients are the same as those of the standard periodic homogenization, and the effective potential is given by the mean. We show that in the short range correlation setting, the limiting distribution of the random part of the homogenization error, as random elements in proper Hilbert spaces, is Gaussian and can be characterized by t…
We consider homogenization problems for first order Hamilton-Jacobi equations with $u^ε/ε$ periodic dependence, recently introduced by C. Imbert and R. Monneau, and also studied by G. Barles: this unusual dependence leads to a nonstandard cell problems. We study the rate of convergence of the solution to the solution of the homogenized problem when the parameter $ε$ tends to 0. We obtain the same rates as those obtained by I. Capuzzo Dolcetta and H. Ishii for the more usual homogenization problems without the dependence in $u^ε/ε$. In a second part, we study Eulerian schemes for the approximat…
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 $\varepsilon$, convergence of the full (variational) spectrum together whit an explicit in $k$ and in $\varepsilon$ order of convergence.
This paper is devoted to establish continuous dependence estimates for the ergodic problem for Bellman operators (namely, estimates of (v_1-v_2) where v_1 and v_2 solve two equations with different coefficients). We shall obtain an estimate of ||v_1-v_2||_\infty with an explicit dependence on the L^\infty-distance between the coefficients and an explicit characterization of the constants and also, under some regularity conditions, an estimate of ||v_1-v_2||_{C^2(\R^n)}. Afterwards, the former result will be crucial in the estimate of the rate of convergence for the homogenization of Bellman eq…
In this paper we prove convergence results for homogenization problem for solutions of partial differential system with rapidly oscillating Dirichlet data. Our method is based on analysis of oscillatory integrals. In the uniformly convex and smooth domain, and smooth operator and boundary data, we prove pointwise convergence results, namely $$|u_{\e}(x)-u_0 (x)| \leq C_κ \e^{(d-1)/2}\frac{1}{d(x)^κ}, \ \forall x\in D, \ \forall \ κ>d-1,$$ where $u_{\e}$ and $u_0$ are solutions of respectively oscillating and homogenized Dirichlet problems, and $d(x)$ is the distance of $x$ from the boundary of…
This paper is devoted to the homogenization of Shrödinger type equations with periodically oscillating coefficients of the diffusion term, and a rapidly oscillating periodic time-dependent potential. One convergence theorem is proved and we derive the macroscopic homogenized model. Our approach is the well known two-scale convergence method.
In this paper we prove convergence results for the homogenization of the Dirichlet problem with rapidly oscillating boundary data in convex polygonal domains. Our analysis is based on integral representation of solutions. Under a certain Diophantine condition on the boundary of the domain and smooth coefficients we prove pointwise, as well as $L^p$ convergence results. For larger exponents $p$ we prove that the $L^p$ convergence rate is close to optimal. We shall also suggest several directions of possible generalization of the result in this paper.
We consider the homogenization to the Brinkman equations for the incompressible Stokes equations in a bounded domain which is perforated by a random collection of small spherical holes. This problem has been studied by the same authors in [A. Giunti and R.M. Höfer, Homogenization for the Stokes equations in randomly perforated domains under almost minimal assumptions on the size of the holes] where convergence of the fluid velocity field towards the solution of the Brinkman equations has been established. In the present we consider the pressure associated to the solution of the Stokes equation…
We revisit the homogenization problem for the Poisson equation in periodically perforated domains with zero Neumann data at the boundary of the holes and prescribed Dirichlet data at the outer boundary. It is known that, if the periodicity of the holes goes to zero but their volume fraction remains fixed and positive, the limit problem is a Dirichlet boundary value problem posed in the domain without the holes, and the effective diffusion coefficients are non-trivially modified; if that volume fraction goes to zero instead, i.e. the holes are dilute, the effective operator remains the Laplacia…
The introduced notion of locally-periodic two-scale convergence allows to average a wider range of microstructures, compared to the periodic one. The compactness theorem for the locally-periodic two-scale convergence and the characterisation of the limit for a sequence bounded in $H^1(Ω)$ are proven. The underlying analysis comprises the approximation of functions, which periodicity with respect to the fast variable depends on the slow variable, by locally-periodic functions, periodic in subdomains smaller than the considered domain, but larger than the size of microscopic structures. The deve…
This paper concentrates on the quantitative homogenization of higher-order elliptic systems with almost-periodic coefficients in bounded Lipschitz domains. For coefficients which are almost-periodic in the sense of H. Weyl, we establish uniform ocal $L^2$ estimates for the approximate correctors. Under an additional assumption on the frequencies of the coefficients (see (1.10)), we derive the existence of the true correctors as well as the sharp $O(\varepsilon)$ convergence rate in $H^{m-1}$. As a byproduct, the large-scale Hölder estimate and a Liouville theorem are obtained for higher-order …
