Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Some of the greatest scientists including Poisson, Faraday, Maxwell, Rayleigh, and Einstein have contributed to the theory of composite materials Mathematically, it is the study of partial differential equations with rapid oscillations in their coefficients Although extensively studied for more than a hundred years, an explosion of ideas in the last five decades (and particularly in the last three decades) has dramatically increased our understanding of the relationship between the properties of the constituent materials, the underlying microstructure of a composite, and the overall effective (electrical, thermal, elastic) moduli which govern the macroscopic behavior This renaissance has been fueled by the technological need for improving our knowledge base of composites, by the advance of the underlying mathematical theory of homogenization, by the discovery of new variational principles, by the recognition of how important the subject is to solving structural optimization problems, and by the realization of the connection with the mathematical problem of quasiconvexification This 2002 book surveys these exciting developments at the frontier of mathematics

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The Theory of Composites

The heterogeneous multiscale method (HMM), a general framework for designing multiscale algorithms, is reviewed. Emphasis is given to the error analysis that comes naturally with the framework. Examples of finite element and finite difference HMM are presented. Applications to dynamical systems and stochastic simulation algorithms with multiple time scales, spall fracture and heat conduction in microprocessors are discussed.

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The heterogeneous multiscale method

Simulation of the multi-scale convergence in computational homogenization approaches

https://www.math.nyu.edu/faculty/majda/pdfFiles/PhysRep%201999%20Majda%20Kramer.pdf

Simplified models for turbulent diffusion : theory, numerical modelling, and physical phenomena

Stabilization of the solution of the Cauchy problem for parabolic equations

We give an example of an incompressible diffusion equation whose solution is nonunique. It is shown that this equation has an approximation solution as well as another solution that cannot be obtained by approximation. We give sufficient conditions for the uniqueness of a solution as well as for the uniqueness of an approximation solution.

Remarks on the Uniqueness of a Solution of the Dirichlet Problem for Second-Order Elliptic Equations with Lower-Order Terms

We study the solvability of an initial-boundary value problem for second-order parabolic equations with variable order of nonlinearity. In the model case, the equation contains the p-Laplacian with a variable exponent p(x, t). We prove that if the measurable exponent p is separated from unity and infinity, then the problem has W-and H-solutions.

Existence theorems for solutions of parabolic equations with variable order of nonlinearity

The paper is concerned with the solvability of the initial-boundary value problem for second-order parabolic equations with variable nonlinearity exponents. In the model case, this equation contains the -Laplacian with a variable exponent . The problem is shown to be uniquely solvable, provided the exponent is bounded away from both 1 and and is log-Holder continuous, and its solution satisfies the energy equality. Bibliography: 18 titles.

Existence and uniqueness theorems for solutions of parabolic equations with a?variable nonlinearity exponent

In this paper, we prove the Holder continuity of solutions of parabolic equations containing the p(x, t)-Laplacian. The degree p must satisfy the so-called logarithmic condition.

V. V. Zhikov

Papers

Stabilization of the solution of the Cauchy problem for parabolic equations

Remarks on the Uniqueness of a Solution of the Dirichlet Problem for Second-Order Elliptic Equations with Lower-Order Terms

Existence theorems for solutions of parabolic equations with variable order of nonlinearity

Existence and uniqueness theorems for solutions of parabolic equations with a?variable nonlinearity exponent

Hölder continuity of solutions of parabolic equations with variable nonlinearity exponent