We consider the homogenization of a non-stationary convection–diffusion equation posed in a bounded domain with periodically oscillating coefficients and homogeneous Dirichlet boundary conditions. Assuming that the convection term is large, we give the asymptotic profile of the solution and determine its rate of decay. In particular, it allows us to characterize the “hot spot”, i.e., the precise asymptotic location of the solution maximum which lies close to the domain boundary and is also the point of concentration. Due to the competition between convection and diffusion, the position of the “hot spot” is not always intuitive as exemplified in some numerical tests.

The paper deals with the homogenization of a non-stationary convection-diffusion equation defined in a thin rod or in a layer with Dirichlet boundary condition. Under the assumption that the convection term is large, we describe the evolution of the solution’s profile and determine the rate of its decay. The main feature of our analysis is that we make no assumption on the support of the initial data which may touch the domain’s boundary. This requires the construction of boundary layer correctors in the homogenization process which, surprisingly, play a crucial role in the definition of the leading order term at the limit. Therefore we have to restrict our attention to simple geometries like a rod or a layer for which the definition of boundary layers is easy and explicit.

We consider the homogenization of a singularly perturbed self-adjoint fourth order elliptic operator with locally periodic coefficients, stated in a bounded domain. We impose Dirichlet boundary conditions on the boundary of the domain. The presence of large parameters in the lower order terms and the dependence of the coefficients on the slow variable lead to localization of the eigenfunctions. We show that the jth eigenfunction can be approximated by a rescaled function that is constructed in terms of the jth eigenfunction of fourth or second order effective operators with constant coefficients.

This paper is aimed at homogenization of an elliptic spectral problem stated in a perforated domain, Fourier boundary conditions being imposed on the boundary of perforation. The presence of a locally periodic coefficient in the boundary operator gives rise to the effect of localization of the eigenfunctions. Moreover, the limit behavior of the lower part of the spectrum can be described in terms of an auxiliary harmonic oscillator operator. We describe the asymptotics of the eigenpairs and derive estimates for the rate of convergence. 

We derive a one-dimensional cable model for the electric potential propagation along an axon. Since the typical thickness of an axon is much smaller than its length, and the myelin sheath is distributed periodically along the neuron, we simplify the problem geometry to a thin cylinder with alternating myelinated and unmyelinated parts. Both the microstructure period and the cylinder thickness are assumed to be of order ε, a small positive parameter. Assuming a nonzero conductivity of the myelin sheath, we find a critical scaling with respect to ε which leads to the appearance of an additional potential in the homogenized nonlinear cable equation. This potential contains information about the geometry of the myelin sheath in the original three-dimensional model. 

We consider a three-dimensional model for a myelinated neuron, which includes Hodgkin–Huxley ordinary differential equations to represent membrane dynamics at Ranvier nodes (unmyelinated areas). Assuming a periodic microstructure with alternating myelinated and unmyelinated parts, we use homogenization methods to derive a one-dimensional nonlinear cable equation describing the potential propagation along the neuron. Since the resistivity of intracellular and extracellular domains is much smaller than the myelin resistivity, we assume this last one to be a perfect insulator and impose homogeneous Neumann boundary conditions on the myelin boundary. In contrast to the case when the conductivity of the myelin is nonzero, no additional terms appear in the one-dimensional limit equation, and the model geometry affects the limit solution implicitly through an auxiliary cell problem used to compute the effective coefficient. We present numerical examples revealing the forecasted dependence of the effective coefficient on the size of the Ranvier node

The paper deals with the asymptotic behaviour of spectra of second order self-adjoint elliptic operators with periodic rapidly oscillating coefficients in the case when the density function (the factor on the spectral parameter) changes sign. We study the Dirichlet problem in a regular bounded domain and show that the spectrum of this problem is discrete and consists of two series, one of them tending towards +∞ and another towards −∞. The asymptotic behaviour of positive and negative eigenvalues and their corresponding eigenfunctions depends crucially on whether the average of the weight function is positive, negative or equal to zero. We construct the asymptotics of eigenpairs in all three cases.

This chapter is devoted to the homogenization of a stationary convection diffusion model problem in a thin rod structure. More precisely, we study the asymptotic behavior of solutions to a boundary value problem for a convection diffusion equation defined in a thin cylinder that is the union of two nonintersecting cylinders with a junction at the origin. We suppose that in each of these cylinders the coefficients are rapidly oscillating functions that are periodic in the axial direction, and that the microstructure period is of the same order as the cylinder diameter. On the lateral boundary of the cylinder we assume the Neumann boundary condition, while at the cylinder bases the Dirichlet boundary conditions are posed.

We consider the homogenization of an elliptic spectral problem with a large potential stated in a thin cylinder with a locally periodic perforation. The size of the perforation gradually varies from point to point. We impose homogeneous Neumann boundary conditions on the boundary of perforation and on the lateral boundary of the cylinder. The presence of a large parameter 1/ε in front of the potential and the dependence of the perforation on the slow variable give rise to the effect of localization of the eigenfunctions. We show that the jth eigenfunction can be approximated by a scaled exponentially decaying function that is constructed in terms of the jth eigenfunction of a one-dimensional harmonic oscillator operator.

The paper deals with homogenization of a spectral problem for a second order self-adjoint elliptic operator stated in a thin cylinder with homogeneous Neumann boundary condition on the lateral boundary and Dirichlet condition on the bases of the cylinder. We assume that the operator coefficients and the spectral density function are locally periodic in the axial direction of the cylinder, and that the spectral density function changes sign. We show that the behavior of the spectrum depends essentially on whether the average of the density function is zero or not. In both cases we construct the effective 1-dimensional spectral problem and prove the convergence of spectra.

The paper deals with a periodic homogenization problem for a non-stationary convection-diffusion equation stated in a thin infinite cylindrical domain with homogeneous Neumann boundary condition on the lateral boundary. It is shown that homogenization result holds in moving coordinates, and that the solution admits an asymptotic expansion which consists of the interior expansion being regular in time, and an initial layer.

The work focuses on the behaviour at infinity of solutions to second order elliptic equation with first order terms in a semi-infinite cylinder. Neumann's boundary condition is imposed on the lateral boundary of the cylinder and Dirichlet condition on its base. Under the assumption that the coefficients stabilize to a periodic regime, we prove the existence of a bounded solution, its stabilization to a constant, and provide necessary and sufficient condition for the uniqueness.

The aim of this paper is to adapt the notion of two-scale convergence in Lp to the case of a measure converging to a singular one. We present a specific case when a thin cylinder with locally periodic rapidly oscillating boundary shrinks to a segment, and the corresponding measure charging the cylinder converges to a one-dimensional Lebegues measure of an interval. The method is then applied to the asymptotic analysis of linear elliptic operators with locally periodic coefficients and a p-Laplacian stated in thin cylinders with locally periodic rapidly varying thickness.

We study the existence and uniqueness of a solution to a linear stationary convection–diffusion equation stated in an infinite cylinder, Neumann boundary condition being imposed on the boundary. We assume that the cylinder is a junction of two semi-infinite cylinders with two different periodic regimes. Depending on the direction of the effective convection in the two semi-infinite cylinders, we either get a unique solution, or one-parameter family of solutions, or even non-existence in the general case. In the latter case we provide necessary and sufficient conditions for the existence of a solution.