Homogenization in L

Homogenization of deterministic control problems with L^∞ running cost is studied by viscosity solutions techniques. It is proved that the value function of an L^∞ problem in a medium with a periodic micro-structure converges uniformly on the compact sets to the value function of the homogenized problem as the period shrinks to 0. Our main convergence result extends that of Ishii (Stochastic Analysis, control, optimization and applications, pp. 305-324, Birkhäuser Boston, Boston, MA, 1999.) to the case of a discontinuous Hamiltonian. The cell problem is solved, but, as non-uniqueness occurs, the effective Hamiltonian must be selected in a careful way. The paper also provides a representation formula for the effective Hamiltonian and gives illustrations to calculus of variations, averaging and one-dimensional problems.     

Convergence of solutions for a network approximation of the two-dimensional Laplace equation in a domain with a system of absolutely conducting disks

A boundary value problem for the Laplace equation describing the (electric, thermal, etc.) field of a system of ideally conducting disks of radius R is considered. The solution to the problem is analyzed under the condition that the characteristic distance δ between the disks is small. It was previously proved that the original continuous problem can be approximated as δ → 0 by a finite-dimensional network problem in the sense that the effective conductivities (energies) of the continuous problem are close to those of its network model. It is shown that the potentials of the ideally conducting disks determined from the continuous problem and the network model are also close to each other as δ → 0, and the difference between the potentials is O(ε^1/4), where ε = δ/R is the characteristic relative distance between the disks.     

A Trotter-Kato type result for a second order difference inclusion in a Hilbert space

A Trotter-Kato type result is proved for a class of second order difference inclusions in a real Hilbert space. The equation contains a nonhomogeneous term f and is governed by a nonlinear operator A, which is supposed to be maximal monotone and strongly monotone. The associated boundary conditions are also of monotone type. One shows that, if An is a sequence of operators which converges to A in the sense of resolvent and fn converges to f in a weighted l²-space, then under additional hypotheses, the sequence of the solutions of the difference inclusion associated to An and fn is uniformly convergent to the solution of the original problem.     

The stressed state of a transversally isotropic medium with a spheroidal inclusion under nonideal mechanical contact

We study the problem of the stressed state of a transversally isotropic medium containing a foreign inclusion in the form of a prolate spheroid under an arbitrary homogeneous stress field at infinity. On the “medium-inclusion” interface there is slipping without flaking. The stressed state is constructed in the medium and in the inclusion using the exterior and interior problems for a prolate spheroid on the basis of potential functions. The solution of the problem is reduced to studying infinite systems of linear algebraic equations. The results of numerical studies are shown as graphs that describe the stress distribution in both the transversally isotropic medium and in the inclusion under various boundary conditions. Four figures. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995, pp. 15–26.     

On the cauchy problem for harmonic maps defined on two-dimensional Minkowski space

Let R¹⁺¹ be two-dimensional Minkowski space and M a complete Riemannian manifold of dimension n. It is proved that the solution of the Cauchy problem for the harmonic map ? : R¹⁺¹ → M exists globally. As an application to physics we conclude that the field function in a two-dimensional chiral field theory is regular for all time, if it is regular initially.     

Non‐local approach in mathematical problems of fluid–structure interaction

Three‐dimensional mathematical problems of interaction between elastic and scalar oscillation fields are investigated. An elastic field is to be defined in a bounded inhomogeneous anisotropic body occupying the domain Ω¯₁⊂ℝ³ while a physical (acoustic) scalar field is to be defined in the exterior domain Ω¯₂=ℝ³\Ω₁ which is filled up also by an anisotropic (fluid) medium. These two fields satisfy the governing equations of steady‐state oscillations in the corresponding domains together with special kinematic and dynamic transmission conditions on the interface ∂Ω₁. The problems are studied by the so‐called non‐local approach, which is the coupling of the boundary integral equation method (in the unbounded domain) and the functional‐variational method (in the bounded domain). The uniqueness and existence theorems are proved and the regularity of solutions are established with the help of the corresponding Steklov–Poincaré type operators and on the basis of the Gårding inequality and the Lax–Milgram theorem. In particular, it is shown that the physical fluid–solid acoustic interaction problem is solvable for arbitrary values of the frequency parameter. Copyright © 1999 John Wiley & Sons, Ltd.     

带参数的一类广义隐拟变分包含问题

研究了一类带参数的广义隐拟变分包含问题,证明了这类问题解的存在性定理,并分析了解的性状和灵敏性.推广了某些相关结果.     

Electromagnetic Scattering by Periodic Structures

This paper is devoted to the scattering of electromagnetic waves by quite general biperiodic structures which may consist of anisotropic optical materials and separate two regions with constant dielectric coefficients. The time-harmonic Maxwell equations are transformed to an equivalent H ¹-variational problem for the magnetic field in a bounded biperiodic cell with nonlocal boundary conditions. The existence of solutions is shown for all physically relevant material parameters. The uniqueness is proved for all frequencies excluding possibly a discrete set. The results of the general problem are compared with known results for a special case, the conical diffraction.     

The direct and inverse problem for an inclusion within a heat-conducting layered medium

This paper is concerned with the problem of heat conduction from an inclusion in a heat transfer layered medium. Making use of the boundary integral equation method, the well-posedness of the forward problem is established by the Fredholm theory. Then an inverse boundary value problem, i.e. identifying the inclusion from the measurements of the temperature and heat flux on the accessible exterior boundary of the medium is considered in the framework of the linear sampling method. Based on a careful analysis of the Dirichlet-to-Neumann map, the mathematical fundamentals of the linear sampling method for reconstructing the inclusion are proved rigorously.     

Regularity results and optimal control problems for the perturbation of boussinesq equations of the ocean

We study in this article a method which computes the variability of current, density and pressure in an oceanic domain. The equations are of Navier-Stokes type for the velocity and pressure, of transport-diffusion type for the density. They are linearized around a given mean circulation and modified by the Boussinesq approximation: density variations are neglected except in the terms of gravity acceleration. The existence and uniqueness of a solution are proved for two sets of equations: first the three-dimensional problem and then the two-dimensional cyclic problem derived by assuming a sinusoidal x-dependence for the perturbation of mean flow. The latter corresponds to a modelization of tropical instability waves which are illustrated by the El Nino phenomenon.

The value of the pressure p on the surface of ocean is of great interest for physical interpretation. To define that quantity, it is necessary to have the regularity p ? H ¹. We have proved that the perturbation (u,ρ,p) of mean circulation is such that: u ? L ²(0T,H ²), ρ ? L ²(0,T H ²) and p ? L ² L ²(0,T H ¹), provided the perturbation of the windstress is sufficiently regular and satisfies compatibility relations. It is proved by means of an extension method, with even-odd reflection. We then develop a problem of control. The observation is the Variability of pressure on the surface of ocean. The control is the variability of windstress f, which acts as to forcing of the perturbation. We prove the existence and uniqueness of an optimal control, which is characterized by a set of equations including the direct problem and the adjoint problem. These results are valid for the three-dimensional problem and the two-dimensional cyclic problem.     

Three-step iterative methods for general variational inclusions in L P spaces

In this paper, we show that the general variational inclusions are equivalent to the fixed point problem. We use this equivalence to discuss the existence of the variational inclusions in L ^p spaces. Using the technique of the updating solution, we suggest some three-step iterative methods for solving the general variational inclusion. We also consider the convergence analysis of the proposed iterative methods under some mild conditions. Since the general variational inclusions include several classes of variational inequalities and optimization problems as special cases, results proved in this paper continue to hold for these problems.     

Periodic solutions of elasticity and electroelasticity theories for a cylinder in R3

General representations are constructed for periodic solutions of the theories of elasticity and electroelasticity for a cylinder in R³. These solutions are utilized to reduce the first boundary value problem in an unbounded isotropic medium weakened by tunnel slit cavities to a system of singular integral equations. The solvability of the characteristic part of the obtained system is proved.Sumy. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 13–20, 1990.     

Convergence of the "predictor — corrector" difference scheme for a first-order linear hyperbolic system of equations with variable coefficients

A new two-step difference scheme is considered for the problem of propagation of acoustic oscillations in a nonhomogeneous moving medium described by a linear hyperbolic system of equations with variable coefficients. Convergence of the difference scheme is proved to second order in_T and h in the L₂ grid norm for solutions from C^3,3 (Q_T).Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, pp. 16–25, 1989.     

Identification of the Time Derivative Coefficient in a Fast Diffusion Degenerate Equation

In this paper, we deal with the identification of the space variable time derivative coefficient u in a degenerate fast diffusion differential inclusion. The function u is vanishing on a subset strictly included in the space domain Ω. This problem is approached as a control problem (P) with the control u. An approximating control problem (P _ε ) is introduced and the existence of an optimal pair is proved. Under certain assumptions on the initial data, the control is found in W ^2,m (Ω), with m>N, in an implicit variational form. Next, it is shown that a sequence of optimal pairs (u_e^*,y_e^*)(u_{\varepsilon }^{\ast },y_{\varepsilon }^{\ast }) of (P _ε ) converges as ε goes to 0 to a pair (u ^*,y ^*) which realizes the minimum in (P), and y ^* is the solution to the original state system.     

Entropy solutions to the Verigin ultraparabolic problem

We study the Cauchy problem for the two-dimensional ultraparabolic model of filtration of a viscous incompressible fluid containing an admixture, with diffusion of the admixture in a porous medium taken into account. The porous medium consists of the fibers directed along some vector field n ^⊥. We prove that if the nonlinearity in the equations of the model and the geometric structure of fibers satisfy some additional “genuine nonlinearity” condition then the Cauchy problem with bounded initial data has at least one entropy solution and the fast oscillating regimes possible in the initial data are promptly suppressed in the entropy solutions. The proofs base on the introduction and systematic study of the kinetic equation associated with the problem as well as on application of the modification of Tartar H-measures which was proposed by Panov.     

退化抛物型方程弱解的存在性

讨论初值为u_0,v_0∈L_+~4(Ω),w∈W~(1,p)(Ω)(p≥2)时退化抛物型方程弱解的存在性.首先利用截断的方法将原问题正则化,化为u_0,v_0∈L_+~∞(Ω)的退化问题,接着对正则化问题的解做估计(这里的估计与具体的截断无关),最后利用弱收敛性,通过取极限的方法证明了原问题解的存在性.     

On the mechanics of helical flows in an ideal incompressible nonviscous continuous medium

We find a general solution to the problem on the motion in an incompressible continuous medium occupying at any time a whole domain D ? R ³ under the conditions that D is an axially symmetric cylinder and the motion is described by the Euler equation together with the continuity equation for an incompressible medium and belongs to the class of helical flows (according to I.S. Gromeka’s terminology), in which sreamlines coincide with vortex lines. This class is constructed by the method of transformation of the geometric structure of a vector field. The solution is characterized in Theorem 2 in the end of the paper.     

The solvability conditions for inverse eigenproblem of symmetric and anti‐persymmetric matrices and its approximation

The problem of generating a matrix A with specified eigen‐pair, where A is a symmetric and anti‐persymmetric matrix, is presented. An existence theorem is given and proved. A general expression of such a matrix is provided. We denote the set of such matrices by ??????_Eⁿ. The optimal approximation problem associated with ??????_Eⁿ is discussed, that is: to find the nearest matrix to a given matrix A* by A∈??????_Eⁿ. The existence and uniqueness of the optimal approximation problem is proved and the expression is provided for this nearest matrix. Copyright © 2002 John Wiley & Sons, Ltd.     

A Posteriori Error Estimates for the Approximations of the Stresses in the Hencky Plasticity Problem

In this article, we derive a posteriori error estimates for the Hencky plasticity problem. These estimates are formulated in terms of the stresses and present guaranteed and computable bounds of the difference between the exact stress field and any approximation of it from the energy space of the dual variational problem. They consist of quantities that can be considered as penalties for the violations of the equilibrium equations, the yield condition and the constitutive relations that must hold for the exact stresses and strains. It is proved that the upper bound tends to zero for any sequence of stresses that tends to the exact solution of the Haar–Karman variational problem. An important ingredient of our analysis is a collection of Poincaré type inequalities involving the L ¹ norms of the tensors of small deformation. Estimates of this form are not new, however we will present computable upper bounds for the constants being involved even for rather complicated domains.     

ΓD‐convergence for a class of quasilinear elliptic equations in thin structures

We study the asymptotic behaviour of the solution of a stationary quasilinear elliptic problem posed in a domain Ω^(ε) of asymptotically degenerating measure, i.e. meas Ω^(ε) → 0 as ε → 0, where ε is the parameter that characterizes the scale of the microstructure. We obtain the convergence of the solution and the homogenized model of the problem is constructed using the notion of convergence in domains of degenerating measure. Proofs are given using the method of local characteristics of the medium Ω^(ε) associated with our problem in a variational form. Copyright © 2005 John Wiley & Sons, Ltd.