In this book, Professor Oleinik highlights her work in the area of partial differential equations. The book is divided into two parts: the first is devoted to the study of the asymptotic behavior at infinity of solutions of a class of nonlinear second order elliptic equations in unbounded and, in particular, cylindrical domains. The second contains the most recent results of the author in the theory of homogenization of partial differential equations and is concerned with questions about partially perforated domains and of solutions with rapidly alternating types of boundary conditions. Many...

1.1. Nearly Integrable Hamiltonian Systems. In this work we examine the system of Hamiltonian equations i = _ iJH, = iJH iJcp iJl with the Hamiltonian function H = Ho(l) ] eH. (I. cp). (1.1) where E: 1 is a small parameter, the perturbation E: Hl (I, cp) is 2n- periodic in CP=CP1,"'CPS' and I is an s-dimensional vector, I = Il, --- I s The CPi are called angular variables, and the Ii action variables. A system with a Hamiltonian depending only on the action variables is said to be integrable, and a system with Hamiltonian (1.1) is said to be nearly integrable. The system (1.1) is also called...

Second order equations with nonnegative characteristic form constitute a new branch of the theory of partial differential equations, having arisen within the last 20 years, and having undergone a particularly intensive development in recent years. An equation of the form (1) is termed an equation of second order with nonnegative characteristic form on a set G, kj if at each point x belonging to G we have a (xHk j 0 for any vector = ( l' ... ' m)' In equation (1) it is assumed that repeated indices are summed from 1 to m, and x = (x l' ---, x ). Such equations are sometimes also called...