Homogenization of a boundary value problem in a thick cascade junction

We consider a homogenization problem in a singularly perturbed two-dimensional domain of a new type that consists of a junction body and many alternating thin rods of two classes. One of the classes consists of rods of finite length, whereas the other contains rods of small length, and inhomogeneous Fourier boundary conditions (the third type boundary conditions) with perturbed coefficients are imposed on boundaries of thin rods. Homogenization theorems are proved. Bibliography: 38 titles. Illustrations: 2 figures. __________ Translated from Problemy Matematicheskogo Analiza, No. 37, 2008, pp. 47–72.     

Homogenization of the Signorini boundary‐value problem in a thick junction and boundary integral equations for the homogenized problem

We consider a mixed boundary‐value problem for the Poisson equation in a thick junction Ω_ε which is the union of a domain Ω₀ and a large number of ε—periodically situated thin cylinders. The non‐uniform Signorini conditions are given on the lateral surfaces of the cylinders. The asymptotic analysis of this problem is done as ε→0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove a convergence theorem and show that the non‐uniform Signorini boundary conditions are transformed in the limiting variational inequalities in the region that is filled up by the thin cylinders as ε→0. The convergence of the energy integrals is proved as well. The existence and uniqueness of the solution to this non‐standard limit problem is established. This solution can be constructed by using a penalty formulation and successive iteration. For some subclass, these problems can be reduced to an obstacle problem in Ω₀ and an appropriate postprocessing. The equations in Ω₀ finally are also treated with boundary integral equations. Copyright © 2010 John Wiley & Sons, Ltd.     

Homogenization of a boundary-value problem with a nonlinear boundary condition in a thick junction of type 3:2:1

We consider a boundary-value problem for the Poisson equation in a thick junction Ω_ε, which is the union of a domain Ω₀ and a large number of ε-periodically situated thin curvilinear cylinders. The following nonlinear Robin boundary condition ∂_νu_ε + εκ(u_ε)=0 is given on the lateral surfaces of the thin cylinders. The asymptotic analysis of this problem is performed as ε → 0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove the convergence theorem and show that the nonlinear Robin boundary condition is transformed (as ε → 0) in the blow-up term of the corresponding ordinary differential equation in the region that is filled up by the thin cylinders in the limit passage. The convergence of the energy integral is proved as well. Using the method of matched asymptotic expansions, the approximation for the solution is constructed and the corresponding asymptotic error estimate in the Sobolev space H¹(Ω_ε) is proved. Copyright © 2007 John Wiley & Sons, Ltd.     

Simple layer potential and the first boundary value problem for a parabolic system on the plane

By the method of boundary integral equations, we construct a classical solution of the first initial–boundary value problem for a one-dimensional (with respect to x) parabolic system in a domain with nonsmooth lateral boundary for the case in which the right-hand sides of the boundary conditions only have continuous derivatives of order 1/2. We study the smoothness of the solution.     

Nonexistence for a boundary value problem arising in parabolic theory

The boundary value problemc _t=c _xx−c _yy+q(t,x)c with {fx349-1} was solved by Colton [1] forq analytic int. The solution may be used for mapping solutions of the heat equation into solutions ofu _t=u _xx+q(t,x)u. Solutions (of the boundary value problem) no longer exist ifq is not analytic int. Erica and Ludwig Jesselson Professor of Theoretical Mathematics, The Weizmann Institute of Science. This research was partially supported by the Minerva Foundation.     

Classical solution of a one-dimensional parabolic boundary value problem with nonlinear boundary conditions and moving boundary

We consider a nonlinear parabolic boundary value problem of the Stefan type with one space variable, which generalizes the model of hydride formation under constant conditions. We suggest a grid method for constructing approximations to the unknown boundary and to the concentration distribution. We prove the uniform convergence of the interpolation approximations to a classical solution of the boundary value problem. (The boundary is smooth, and the concentration distribution has the necessary derivatives.) Thus, we prove the theorem on the existence of a solution, and the proof is given in constructive form: the suggested convergent grid method can be used for numerical experiments.     

Maximal space regularity in nonhoomogeneous initial boundary value parabolic problem

Maximal regularity results for second order linear parabolic nonhoomogeneous initial-boundary value problems are established. They are used to show existence, uniqueness and C¹ dependence on the initial value of the solution of general fully nonlinear problems.     

Homogenization limit of a parabolic equation with nonlinear boundary conditions

Consider the parabolic equation

(E)     

On the blowing-up behavior of solutions for a parabolic boundary value problem

This paper discusses the existence and the blowing-up behaviour of the solution for an initial boundary value problem which arises from the ignition of mixtures of gases. It is shown under the Dirichlet or the third type of boundary condition that for certain a class of initial functions local solutions exist and grow unbounded in finite time, while for another class of initial functions there exist global solutions which converge to a steady state solution of the problem. These results lead to an interesting bifurcation phenomenon on the existence, stability and blowing-up property of the solution in terms of either the strength of the nonlinear function or the size of the diffusion region. Estimates for the stability and instability regions as well as bounds for the finite escape time are explicitly given.     

On a first boundary value problem for hyperbolic equations in the plane

In the paper we study a boundary value problem for a hyperbolic equation with two independent variables; this problem is a generalization of the well-known Darboux problem. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 294–306, February, 1999.