Fabio Camilli, Olivier Ley, Paola Loreti
In this paper we study homogenization for a class of monotone systems of first-order time-dependent periodic Hamilton-Jacobi equations. We characterize the Hamiltonians of the limit problem by appropriate cell problems. Hence we show the uniform convergence of the solution of the oscillating systems to the bounded uniformly continuous solution of the homogenized system.
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In the homogenization of monotone parabolic partial differential equations with oscillations in both the space and time variables the gradients converges only weakly in $L^p$. In the present paper we construct a family of correctors, such that, up to a remainder which converges to zero strongly in $L^p$, we obtain strong convergence of the gradients in $L^p$.
In this work we deal with the stochastic homogenization of the initial boundary value problems of monotone type. The models of monotone type under consideration describe the deformation behaviour of inelastic materials with a microstructure which can be characterised by random measures. Based on the Fitzpatrick function concept we reduce the study of the asymptotic behaviour of monotone operators associated with our models to the problem of the stochastic homogenization of convex functionals within an ergodic and stationary setting. The concept of Fitzpatrick's function helps us to introduce a…
In this paper we prove a general homogenization result for monotone parabolic problems with an arbitrary number of microscopic scales in space as well as in time, where the scale functions are not necessarily powers of epsilon. The main tools for the homogenization procedure are multiscale convergence and very weak multiscale convergence, both adapted to evo lution problems. At the end of the paper an example is given to concretize the use of the main result.
Aim of this paper is to extend the continuous dependence estimates proved in \cite{JK1} to quasi-monotone systems of fully nonlinear second-order parabolic equations. As by-product of these estimates, we get an Hölder estimate for bounded solutions of systems and a rate of convergence estimate for the vanishing viscosity approximation. In the second part of the paper we employ similar techniques to study the periodic homogenization of quasi-monotone systems of fully nonlinear second-order uniformly parabolic equations. Finally, some examples are discussed.
We consider the homogenization of monotone systems of viscous Hamilton-Jacobi equations with convex nonlinearities set in the stationary, ergodic setting. The primary focus of this paper is on collapsing systems which, as the microscopic scale tends to zero, average to a deterministic scalar Hamilton-Jacobi equation. However, our methods also apply to systems which do not collapse and, as the microscopic scale tends to zero, average to a deterministic system of Hamilton-Jacobi equations.
Reiterated Periodic Homogenization of Parabolic Monotone Operators with Nonstandard Growth
本文研究导致Orlicz空间的非标准增长单调算子抛物问题的重述周期均匀化，通过重述双尺度收敛方法确定全局和宏观均匀化问题。
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.
Homogenization of a nonlinear monotone problem in a locally periodic domain via unfolding method
该研究针对局部周期振荡域上带混合边界条件的非线性单调问题，采用展开方法证明了零延拓解及其流在弱L²意义下收敛到极限分布问题的解，推进了周期域均匀化理论的相关研究。
In this work we study the homogenization for infinitesimal dislocation based gradient viscoplasticity with linear kinematic hardening and general non-associative monotone plastic flows. The constitutive equations in the models we study are assumed to be only of monotone type. Based on the generalized version of Korn's inequality for incompatible tensor fields (the non-symmetric plastic distortion) due to Neff/Pauly/Witsch, we derive uniform estimates for the solutions of quasistatic initial-boundary value problems under consideration and then using an unfolding operator technique and a monoton…
Regularity for monotone Operators and applications to homogenization of $p$-Laplace type equations
本研究针对主部缺乏强单调性的拟线性方程解的梯度，给出局部L^q估计，进而推导退化/奇异拟线性方程解梯度的一致大尺度L^q估计与非退化方程解的大尺度Lipschitz估计，为p-Laplace型方程均匀化提供正则性支撑。
We study the homogenization of a stationary random maximal monotone operator on a probability space equipped with an ergodic dynamical system. The proof relies on Fitzpatrick's variational formulation of monotone relations, on Visintin's scale integration/disintegration theory and on Tartar-Murat's compensated compactness. We provide applications to systems of PDEs with random coefficients arising in electromagnetism and in nonlinear elasticity.
This paper deals with the limit behaviour of the solutions of quasi-linear equations of the form \ $\ds -\limfunc{div}\left(a\left(x, x/{\varepsilon _h},Du_h\right)\right)=f_h$ on $Ω$ with Dirichlet boundary conditions. The sequence $(\varepsilon _h)$ tends to $0$ and the map $a(x,y,ξ)$ is periodic in $y$, monotone in $ξ$ and satisfies suitable continuity conditions. It is proved that $u_h\rightarrow u$ weakly in $H_0^{1,2}(Ω)$, where $u$ is the solution of a homogenized problem \ $-\limfunc{div}(b(x,Du))=f$ on $Ω$. We also prove some corrector results, i.e. we find $(P_h)$ such that $Du_h-P_h…
Explicit corrector in homogenization of monotone operators and its application to nonlinear dielectric elastomer composites
本文研究了非线性静电与电致伸缩弹性弱耦合系统的周期均匀化，证明了有效响应由依赖于原微观结构的均匀介电弹性体描述，并在最小正则性下获得了单调算子均匀化的显式校正子及两个 $L^p-$ 梯度估计。
The paper deals with homogenization problem for nonlinear elliptic and parabolic equations in a periodically perforated domain, a nonlinear Fourier boundary conditions being imposed on the perforation border. Under the assumptions that the studied differential equation satisfies monotonicity and 2-growth conditions and that the coefficient of the boundary operator is centered at each level set of unknown function, we show that the problem under consideration admits homogenization and derive the effective model.
We continue the study of the homogenization of coercive non-convex Hamilton-Jacobi equations in random media identifying two general classes of Hamiltonians with very distinct behavior. For the first class there is no homogenization in a particular environment while for the second homogenization takes place in environments with finite range dependence. Motivated by the recent counter-example of Ziliotto, who constructed a coercive but non-convex Hamilton-Jacobi equation with stationary ergodic random potential field for which homogenization does not hold, we show that same happens for coercive…
We study a cell problem arising in homogenization for a Hamilton-Jacobi equation whose Hamiltonian is not coercive. We introduce a generalized notion of effective Hamiltonians by approximating the equation and characterize the solvability of the cell problem in terms of the generalized effective Hamiltonian. Under some sufficient conditions, the result is applied to the associated homogenization problem. We also show that homogenization for non-coercive equations fails in general.
We provide an example of a Hamilton-Jacobi equation in which stochastic homogenization does not occur. The Hamiltonian involved in this example satisfies the standard assumptions of the literature, except that it is not convex.
We study homogenization for a class of stationnary Hamilton-Jacobi equations in which the Hamiltonian is obtained by perturbing near the origin an otherwise periodic Hamiltonian. We prove that the limiting problem consists of a Hamilton-Jacobi equation outside the origin, with the same effective Hamiltonian as in periodic homogenization, supplemented at the origin with an effective Dirichlet condition that keeps track of the perturbation. Various comments and extensions are discussed.
We provide a general result concerning the homogenization of nonconvex viscous Hamilton-Jacobi equations in the stationary, ergodic setting. In particular, we show that homogenization occurs for a non-empty set of points within every level set of the effective Hamiltonian, and for every point in the minimal level set of the effective Hamiltonian. In addition, these methods provide a new proof of homogenization, in full, for convex equations and, for a class of level-set convex equations. Finally, we prove that the question of homogenization for first order equations reduces to the case that th…
In this paper, we prove the stochastic homogenization of certain nonconvex Hamilton-Jacobi equations. The nonconvex Hamiltonians, which are generally uneven and inseparable, are generated by a sequence of quasiconvex Hamiltonians and a sequence of quasiconcave Hamiltonians through the min-max formula. We provide a monotonicity assumption on the contact values between those stably paired Hamiltonians so as to guarantee the stochastic homogenzation.
In this paper, we will prove the random homogenization of general coercive non-convex Hamilton-Jacobi equations in one dimensional case. This extends the result of Armstrong, Tran and Yu when the Hamiltonian has a separable form $H(p,x,ω)=H(p)+V(x,ω)$ for any coercive $H(p)$.
We study the homogenization of obstacle problems in Orlicz-Sobolev spaces for a wide class of monotone operators (possibly degenerate or singular) of the $p(\cdot)$-Laplacian type. Our approach is based on the Lewy-Stampacchia inequalities, which then give access to a compactness argument. We also prove the convergence of the coincidence sets under non-degeneracy conditions.
We prove, under some assumptions, the existence of correctors for the stochastic homoge-nization of of " viscous " possibly degenerate Hamilton-Jacobi equations in stationary ergodic media. The general claim is that, assuming knowledge of homogenization in probability, correctors exist for all extreme points of the convex hull of the sublevel sets of the effective Hamiltonian. Even when homogenization is not a priori known, the arguments imply existence of correctors and, hence, homogenization in some new settings. These include positively homogeneous Hamiltonians and, hence, geometric-type eq…
Rate of convergence for homogenization of nonlinear weakly coupled Hamilton-Jacobi systems
该研究针对凸情形下非线性弱耦合Hamilton-Jacobi方程组的周期均匀化问题，推导并证明了精确的O(√ε)收敛速率。
We present a simple new proof for the stochastic homogenization of quasiconvex (level-set convex) Hamilton-Jacobi equations set in stationary ergodic environments. Our approach, which is new even in the convex case, yields more information about the qualitative behavior of the effective nonlinearity.
The quasi-periodic homogenization for some classes of first-order Hamilton-Jaconi-Bellman equation is studied in this paper. The cell problem of the quasi-periodic homogenization satisfies the non-resonance condition, under which the corresponding deterministic system is ergodic. The almost periodic homogenization for the same classes of equations is also solved, as a limit of a sequence of quasi-periodic homogenizations. Here, the almost periodicity is in the sense of H. Bohr. This result has been cited by some authors, for example: by H. Ishii, "Almost periodic homogenization of Hamilton-Jac…
We present a proof of qualitative stochastic homogenization for a nonconvex Hamilton-Jacobi equation. The new idea is to introduce a family of "sub-equations" and to control solutions of the original equation by the maximal subsolutions of the latter, which have deterministic limits by the subadditive ergodic theorem and maximality.
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.
We present qualitative and quantitative homogenization results for pathwise Hamilton-Jacobi equations with "rough" multiplicative driving signals. When there is only one such signal and the Hamiltonian is convex, we show that the equation, as well as equations with smooth approximating paths, homogenize. In the multi-signal setting, we demonstrate that blow-up or homogenization may take place. The paper also includes a new well-posedness result, which gives explicit estimates for the continuity of the solution map and the equicontinuity of solutions in the spatial variable.
Homogenization of a relaxed compressible viscous two-phase fluid model in a domain with very tiny holes
该研究针对带极小孔洞区域的可压缩Navier-Stokes-Korteweg近似系统，证明当障碍物尺寸远快于间距趋于零时极限系统不变，扩展了单相可压缩Navier-Stokes方程的均质化结果至两相流体情形。
Quantitative homogenization of state-constraint Hamilton--Jacobi equations on perforated domains and applications
本文研究穿孔区域上状态约束Hamilton--Jacobi方程的周期均匀化，在凸情形下获得最优收敛速率，并进一步处理小孔稀疏情形及部分孔缺失的区域缺陷均匀化问题。
Polynomial Convergence Rate for Quasi-Periodic Homogenization of Hamilton-Jacobi Equations and Application to Ergodic Estimates
本文利用具Diophantine频率有界准周期函数的新定量遍历估计，证明了具准周期势的Hamilton-Jacobi方程均匀化的多项式收敛速率，建立了收敛速率与有效Hamiltonian正则性的联系，并应用于研究无界准周期函数Birkhoff平均的收敛速率。
Quantitative homogenization of convex Hamilton-Jacobi equations with $u/\varepsilon$-periodic Hamiltonians
本文研究了位错动力学中具有 $u/\varepsilon$-周期 Hamiltonian 的一阶凸 Hamilton-Jacobi 方程的定量均匀化，利用基本解和隐式变分原理建立了最优收敛速率，并进一步证明了解和校正子的全局 Hölder 正则性。
We look at the effective Hamiltonian $\bar H$ associated with the Hamiltonian $H(p,x)=H(p)+V(x)$ in the periodic homogenization theory. Our central goal is to understand the relation between $V$ and $\bar H$. We formulate some inverse problems concerning this relation. Such type of inverse problems are in general very challenging. In the paper, we discuss several special cases in both convex and nonconvex settings.
We consider a Hamilton-Jacobi equation where the Hamiltonian is periodic in space and coercive and convex in momentum. Combining the representation formula from optimal control theory and a theorem of Alexander, originally proved in the context of first-passage percolation, we find a rate of homogenization which is within a log-factor of optimal and holds in all dimensions.
We consider the specified stochastic homogenization of first order evolutive Hamilton-Jacobi equations on a very simple junction, i.e the real line with a junction at the origin. Far from the origin, we assume that the considered hamiltonian is closed to given stationary ergodic hamiltonians (which are different on the left and on the right). Near the origin, there is a perturbation zone which allows to pass from one hamiltonian to the other. The main result of this paper is a stochastic homogenization as the length of the transition zone goes to zero. More precisely, at the limit we get two d…
Recently, C. Imbert & R. Monneau study the homogenization of coercive Hamilton-Jacobi Equations with a $u/e$-dependence : this unusual dependence leads to a non-standard cell problem and, in order to solve it, they introduce new ideas to obtain the estimates on the oscillations of the solutions. In this article, we use their ideas to provide new homogenization results for ``standard'' Hamilton-Jacobi Equations (i.e. without a $u/e$-dependence) but in the case of {\it non-coercive Hamiltonians}. As a by-product, we obtain a simpler and more natural proof of the results of C. Imbert & R. Monneau…
Quantitative homogenization of Hamilton--Jacobi equations on perforated domains with Dirichlet boundary conditions
本文研究了带Dirichlet边界条件的穿孔域上凸Hamilton-Jacobi方程的周期均匀化，通过分析解的最优控制表示与度量函数性质，建立了最优收敛速率并处理了相关奇异性问题。
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…
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…
We study the stochastic homogenization for a Cauchy problem for a first-order Hamilton-Jacobi equation whose operator is not coercive w.r.t. the gradient variable. We look at Hamiltonians like $H(x,σ(x)p,ω)$ where $σ(x)$ is a matrix associated to a Carnot group. The rescaling considered is consistent with the underlying Carnot group structure, thus anisotropic. We will prove that under suitable assumptions for the Hamiltonian, the solutions of the $\varepsilon$-problem converge to a deterministic function which can be characterized as the unique (viscosity) solution of a suitable deterministic…
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.
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.
This paper deals with the periodic homogenization of nonlocal parabolic Hamilton-Jacobi equations with superlinear growth in the gradient terms. We show that the problem presents different features depending on the order of the nonlocal operator, giving rise to three different limit problems. To prove the locally uniform convergence to the unique solution of the Cauchy problem for the effective equation we need a new comparison principle among viscosity semi-solutions of integro-differential equations that can be of independent interest.
Homogenization of Viscous Sublinear Hamilton--Jacobi Equations with $u/ε$-Dependence
本文研究含未知函数快速依赖的粘性次线性Hamilton--Jacobi方程的周期均匀化，证明小参数下的定性均匀化结果并建立演化胞腔问题的大时间平均结论。
Quantitative homogenization for obstacle problems of convex Hamilton-Jacobi equations
针对周期介质中依赖双尺度空间变量的凸哈密尔顿-雅可比方程障碍问题，研究者建立了利普希茨障碍下均匀化极限的最优收敛速率O(ε)，并构造例子说明非利普希茨障碍会导致收敛速率恶化。
Quantitative homogenization of convex Hamilton-Jacobi equations with Neumann type boundary conditions
本文研究了凸哈密尔顿-雅可比方程在Neumann型边界条件下的周期均质化问题，通过新表示公式证明了扩展度量函数的子加性和超加性，并获得了最优收敛速率O(ε)。
We consider homogenization for weakly coupled systems of Hamilton--Jacobi equations with fast switching rates. The fast switching rate terms force the solutions converge to the same limit, which is a solution of the effective equation. We discover the appearance of the initial layers, which appear naturally when we consider the systems with different initial data and analyze them rigorously. In particular, we obtain matched asymptotic solutions of the systems and rate of convergence. We also investigate properties of the effective Hamiltonian of weakly coupled systems and show some examples wh…
Periodic Homogenization of Hamilton-Jacobi Equations for Infinite Systems of Indistinguishable Particles
本文研究了在无限维Hilbert空间上Hamilton-Jacobi方程的均质化问题，针对在环面上的无限多个不可区分粒子系统。通过关联的单元问题，作者在合适条件下表征了有效Hamiltonian，并证明了解以$O(\varepsilon^{1/3})$的速度收敛到极限方程的解，从而得到该类可能非凸Hamilton-Jacobi方程在无限维中的定性和定量均质化结果。
Loss of quasiconvexity in the periodic homogenization of viscous Hamilton-Jacobi equations
本文研究一致椭圆 Hamilton–Jacobi 方程周期均匀化中有效 Hamilton 量的拟凸性，证明其可能丧失该性质，并在一维情形说明这种丧失具有普遍性。
We prove that the effective nonlinearities (ergodic constants) obtained in the stochastic homogenization of Hamilton-Jacobi, "viscous" Hamilton-Jacobi and nonlinear uniformly elliptic pde are approximated by the analogous quantities of appropriate "periodizations" of the equations. We also obtain an error estimate, when there is a rate of convergence for the stochastic homogenization.
In this paper we establish periodic homogenization for Hamilton-Jacobi-Bellman (HJB) equations, associated to nonlocal operators of integro-differential type. We consider the case when the fractional diffusion has the same order as the drift term, and is weakly elliptic. The outcome of the paper is twofold. One one hand, we provide Lipschitz regularity results for weakly elliptic non-local HJB, extending the results previously obtained in [8]. On the other hand, we establish a convergence result, based on half relaxed limits and a comparison principle for the effective problem. The latter stro…
Capuzzo-Dolcetta and Ishii proved that the rate of periodic homogenization for coercive Hamilton-Jacobi equations is $O(\varepsilon^{1/3})$. We complement this result by constructing examples of coercive nonconvex Hamiltonians whose rate of periodic homogenization is $Ω(\varepsilon^{1/2})$.
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…
Oscillations in Compressible Navier-Stokes and Homogenization in Phase Transition problems
本文研究了具有非单调压力的可压缩 Navier-Stokes 系统中由初始数据诱导的持续振荡的精确解，并利用守恒律的动理学公式导出了描述粘弹性材料相变的一维模型中振荡传播的有效均匀化方程。
We give an example of the failure of homogenization for a viscous Hamilton-Jacobi equation with non-convex Hamiltonian.
Stochastic homogenization of quasiconvex degenerate viscous HJ equations in 1d
该研究证明了一维平稳遍历环境下拟凸超线性哈密顿量的退化粘性HJ方程的随机均匀化，并首次（含周期情形）发现有效哈密顿量具有拟凸性。
We proof the homogenization of the Hamilton-Jacobi equation on arbitrary compact manifolds using Evans perturbed test function method.
Stochastic Homogenization of HJ Equations: a Differential Game Approach
该研究针对一类可表示为最大-最小公式的非凸非强制一阶HJ方程，提出微分博弈论方法证明其在有限范围依赖环境下的随机均匀化，并推广至一类Lipschitz哈密顿量，同时得到线性初值下解收敛的定量速率。
Let $u^\varepsilon$ and $u$ be viscosity solutions of the oscillatory Hamilton-Jacobi equation and its corresponding effective equation. Given bounded, Lipschitz initial data, we present a simple proof to obtain the optimal rate of convergence $\mathcal{O}(\varepsilon)$ of $u^\varepsilon \rightarrow u$ as $\varepsilon \rightarrow 0^+$ for a large class of convex Hamiltonians $H(x,y,p)$ in one dimension. This class includes the Hamiltonians from classical mechanics with separable potential. The proof makes use of optimal control theory and a quantitative version of the ergodic theorem for perio…
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