Fluctuations in the homogenization of semilinear equations with random potentials

We study the stochastic homogenization and obtain a random fluctuation theory for semilinear elliptic equations with a rapidly varying random potential. To first order, the effective potential is the average potential and the nonlinearity is not affected by the randomness. We then study the limiting distribution of the properly scaled homogenization error (random fluctuations) in the space of square integrable functions, and prove that the limit is a Gaussian distribution characterized by homogenized solution, the Green’s function of the linearized equation around the homogenized solution, and by the integral of the correlation function of the random potential. These results enlarge the scope of the framework that we have developed for linear equations to the class of semilinear equations.     

Boundary homogenization in domains with randomly oscillating boundary

We consider a model homogenization problem for the Poisson equation in a domain with a rapidly oscillating boundary which is a small random perturbation of a fixed hypersurface. A Fourier boundary condition with random coefficients is imposed on the oscillating boundary. We derive the effective boundary condition, prove a convergence result, and establish error estimates.     

Homogenization of parabolic equations with large time-dependent random potential

This paper concerns the homogenization problem of a parabolic equation with large, time-dependent, random potentials in high dimensions d≥3$d\ge 3$. Depending on the competition between temporal and spatial mixing of the randomness, the homogenization procedure turns to be different. We characterize the difference by proving the corresponding weak convergence of Brownian motion in random scenery. When the potential depends on the spatial variable macroscopically, we prove a convergence to SPDE.     

Maximum Principle in the Optimal Design of Plates with Stratified Thickness

An optimal design problem for a plate governed by a linear, elliptic equation with bounded thickness varying only in a single prescribed direction and with unilateral isoperimetrical-type constraints is considered. Using Murat–Tartars homogenization theory for stratified plates and Young-measure relaxation theory, smoothness of the extended cost and constraint functionals is proved, and then the maximum principle necessary for an optimal relaxed design is derived.     

Singular homogenization with stationary in time and periodic in space coefficients

We study the homogenization problem for a random parabolic operator with coefficients rapidly oscillating in both the space and time variables and with a large highly oscillating nonlinear potential, in a general stationary and mixing random media, which is periodic in space. It is shown that a solution of the corresponding Cauchy problem converges in law to a solution of a limit stochastic PDE.     

Homogenization of dynamic laminates

This paper addresses the study of the homogenization problem associated with propagation of long wave disturbances in materials whose properties exhibit not only spacial but also temporal inhomogeneities (called dynamic materials). The study was initiated by Lurie in his pioneering work of 1997. Homogenization theory is employed to replace an equation with oscillating coefficients by a homogenized equation. Two typical examples of periodic homogenization are considered: the wave equation and Maxwell's system coefficients oscillating rapidly not only in space but also in time. Conditions that generate applicability of the homogenization procedure to dynamic materials composites are developed. In particular, we examine a cell problem for periodic composites as well as the laminate formulae. The effective tensors of rank-one laminates for one-dimensional wave equation and the full Maxwell's system are computed explicitly. We also note some dramatic differences between the hyperbolic and the elliptic cases.     

Critical Homogenization of SDEs Driven by a Levy Process in Random Medium

We are concerned with homogenization of stochastic differential equations (SDE) with stationary coefficients driven by Poisson random measures and Brownian motions in the critical case, that is, when the limiting equation admits both a Brownian part as well as a pure jump part. We state an annealed convergence theorem. This problem is deeply connected with homogenization of integral partial differential equations.     

FE heterogeneous multiscale method for long-time wave propagation

A new finite element heterogeneous multiscale method (FE-HMM) is proposed for the numerical solution of the wave equation over long times in a rapidly varying medium. Our FE-HMM captures long-time dispersive effects of the true solution at a cost similar to that of a standard numerical homogenization scheme which, however, only captures the short-time macroscale behavior of the wave field.     

Homogenization of a Nonlinear Convection-Diffusion Equation with Rapidly Oscillating Coefficients and Strong Convection

A Cauchy problem for a nonlinear convection-diffusion equationwith periodic rapidly oscillating coefficients is studied. Underthe assumption that the convection term is large, it is provedthat the limit (homogenized) equation is a nonlinear diffusionequation which shows dispersion effects. The convergence ofthe homogenization procedure is justified by using a new versionof a two-scale convergence technique adapted to rapidly movingcoordinates.     

A new model for nonlinear elastic plates with rapidly varying thickness,ii: the effect of the behavior of the forces when the thickness approaches zero

We consider a family of nonlinear elastic plates with rapidly varying thickness under the assumption that the three-dimensional constitutive equation is linear with respect to the "full" strain tensor (St. Venant-Kirchhoff material). The main goal of this paper is to shown that the limit problem, when the mean plate thickness converges to zero, may be a ill posed problem if the forces do not behave in an appropriate manner     

Fokker-planck equation for brownian motion of rotating spherical particles : PMM vol. 40, n 6, 1976, pp. 1127–1131

The Langevin equation to derive the Fokker-Planck equation is used for the Brownian motion of particles in translational motion. The Fokker-Planck equation for the Brownian motion of particles which have, in addition to the translational velocity also an angular velocity, has not, so far, been derived. This can apparently be explained by the fact that in the case of the rotational motion, the Langevin equation for the translational motion velocity vector must be supplemented by a corresponding equation for an angular velocity vector. The latter equation must contain, in addition to the systematic moment of reaction linearly dependent on the angular velocity of rotation itself, a random moment rapidly varying with time. Moreover, to ensure the compatibility of two differential vector equations within the system, additional relations which must be introduced, must connect not only the coefficients of the systematic reactions, but also the. random vectors varying rapidly with time.In [1],the Boltzmann's equation for a mixture of two gases was used to derive a Fokker-Planck equation for a translational motion of Brownian particles. The same method can be applied to the Brownian motion of spherical particles which have, in addition to the translational velocities, angular velocities of self-rotations. In this case there is no need to introduce additional relations connecting the random rapidly varying vectors.In the present paper we derive the Fokker-Planck equations for a new model of rotating spherical molecules which was used in [2].     

Water waves over a rough bottom in the shallow water regime

This is a study of the Euler equations for free surface water waves in the case of varying bathymetry, considering the problem in the shallow water scaling regime. In the case of rapidly varying periodic bottom boundaries this is a problem of homogenization theory. In this setting we derive a new model system of equations, consisting of the classical shallow water equations coupled with nonlocal evolution equations for a periodic corrector term. We also exhibit a new resonance phenomenon between surface waves and a periodic bottom. This resonance, which gives rise to secular growth of surface wave patterns, can be viewed as a nonlinear generalization of the classical Bragg resonance. We justify the derivation of our model with a rigorous mathematical analysis of the scaling limit and the resulting error terms. The principal issue is that the shallow water limit and the homogenization process must be performed simultaneously. Our model equations and the error analysis are valid for both the two- and the three-dimensional physical problems.     

On the rate of convergence of solutions in domain with periodic multilevel oscillating boundary

In this paper we deal with the homogenization problem for the Poisson equation in a singularly perturbed domain with multilevel periodically oscillating boundary. This domain consists of the body, a large number of thin cylinders joining to the body through the thin transmission zone with rapidly oscillating boundary. Inhomogeneous Fourier boundary conditions with perturbed coefficients are set on the boundaries of the thin cylinders and on the boundary of the transmission zone. We prove the homogenization theorems and derive the estimates for the convergence of the solutions. Copyright © 2010 John Wiley & Sons, Ltd.     

HOMOGENIZATION OF SEMILINEAR PARABOLIC EQUATIONS IN PERFORATED DOMAINS

This paper is devoted to the homogenization of a semilinear parabolic equation with rapidly oscillating coefficients in a domain periodically perforated byε-periodic holes of size ε. A Neumann condition is prescribed on the boundary of the holes.The presence of the holes does not allow to prove a compactness of the solutions in L2. To overcome this difficulty, the authors introduce a suitable auxiliary linear problem to which a corrector result is applied. Then, the asymptotic behaviour of the semilinear problem as ε→ 0 is described, and the limit equation is given.     

Relaxation of Optimal Control Problem Governed by Semilinear Elliptic Equation with Leading Term Containing Controls

An optimal control problem governed by semilinear elliptic partial differential equation is considered. The equation is in divergence form with the leading term containing controls. A relaxed problem is constructed by homogenization. By studying the G-closure problem, a local representation of admissible set of relaxed control is given. Finally, the maximum principle of relaxed problem is established via homogenization spike variation.     

Diffusion in a locally stationary random environment

This paper deals with homogenization of diffusion processes in a locally stationary random environment. Roughly speaking, such an environment possesses two evolution scales: both a fast microscopic one and a smoothly varying macroscopic one. The homogenization procedure aims at giving a macroscopic approximation that takes into account the microscopic heterogeneities.     

Homogenization of “Viscous” Hamilton–Jacobi Equations in Stationary Ergodic Media

ABSTRACT

We study the homogenization of “viscous” Hamilton–Jacobi equations in stationary ergodic media. The “viscosity” and the spatial oscillations are assumed to be of the same order. We identify the asymptotic (effective) equation, which is a first-order deterministic Hamilton–Jacobi equation. We also provide examples that show that the associated macroscopic problem does not admit suitable solutions (correctors). Finally, we present as applications results about large deviations of diffusion processes and front propagation (asymptotics of reaction-diffusion equations) in random environments.     

Homogenization and dispersion effects in the problem of propagation of waves generated by a localized source

We construct asymptotic solutions to the wave equation with velocity rapidly oscillating against a smoothly varying background and with localized initial perturbations. First, using adiabatic approximation in the operator form, we perform homogenization that leads to a linearized Boussinesq-type equation with smooth coefficients and weak “anomalous” dispersion. Then, asymptotic solutions to this and, as a consequence, to the original equations are constructed by means of a modified Maslov canonical operator; for initial perturbations of special form, these solutions are expressed in terms of combinations of products of the Airy functions of a complex argument. On the basis of explicit formulas obtained, we analyze the effect of fast oscillations of the velocity on the solution fronts and solution profiles near the front.     

“Splashes” in Fredholm integro-differential equations with rapidly varying kernels

We consider a singularly perturbed Fredholm integro-differential equation with a rapidly varying kernel. We derive an algorithmfor constructing regularized asymptotic solutions. It is shown that, given a rapidly decreasing multiplier of the kernel, the original problem does no involve the spectrum (i.e., it is solvable for any right-hand side).