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Book review

Author(s) Gindikin, S.
Volevich, L.R.
Title The Method of Newton's Polyhedron in the Theory of Partial Differential Equations
Publisher Kluwer Academic Publishers
Year of publication 1992
Reviewed by P.P. Zabrejko

This book is devoted to the application of Newton's polygon and polyhedron for solving some problems in the theory of partial differential equations. As it is well known Newton's polygon (in two dimensional case) and polyhedron (in three and more dimensional case) of some polynomial (or formal power series) is convex hull of the set of exponents of its nontrivial monomials (sometimes completed in some definite way). The boundary of Newton's polygon or polyhedron can be interpreted as one of the possible generalizations of the degree of polynomial (or the order of the formal power series) of one variable to the general case. It is a much informative notion than that of the ordinary degree or order. At the same time it is not invariant with respect to linear transformations of coordinate systems. In general the Newton's polygon and polyhedron accumulate information about the principal monomials under various weighted degrees of polynomials.

It is well-known that the Newton's polygon and polyhedron play an important role in the theory of small and large solutions of nonlinear operator equations, theory of perturbations of linear operators, the bifurcation and singularities theory, analytical theory of ordinary differential equations and so on. However the applications of Newton's polygon and polyhedron in other branches of analysis are known a few and this book is a pleasant exception It turns out that the Newton's polygon and polyhedron are an important and convinient approach to investigating of some difficult problems in the partial differential equations theory.

The book splits into two parts. The first part (chapters 1-4) deals with Newton's polygon and its applications to the partial differentia] equations with two variables and some special applications to partial differential equations with more than two variables. Chapter 1 "Two-sided estimates for polynomials related to Newton's polygon and their application to studying local properties of partial differential operators in two variables" deals primarily with analysis of inequalities of type <formula>; the Newton's polygon allows effectively to describe the behavior of the considered polynomial in the infinity and then to obtain corresponding estimates of ellipticness for corresponding partial differential equation. In particular in chapter N quasi-elliptic polynomials (hypoelliptic polynomials for which the increasing order in the infinity is defined only with monomials corresponding to vertexes of the Newton's polygon of this polynomial) and corresponding partial differential equations are studied. The chapter 2 "Parabolic operators associated with Newton's polygon" deals with the parabolic due to I.G. Petrovskii partial differential equations in terms of Newton's polygon and some their natural generalizations that are called N-parabolic and N-stable ones; then the correctness of Cauchy problem for N-parabolic and N-stable partial differential equations with constant and variable coefficients is presented; at first the case of two variables is considered and then the general case is described. Chapter 3 "Dominantly correct operators" presents a new natural generalization of correct in Petrovskii's sense partial differential equations - dominantly correct ones that are defined with polynomials for which the minor monomials are estimated by means of the polynomial itself with an arbitrary large constant when the imaginary part of time-variable is sufficiently large. In the chapter a simple algebraic description of dominantly correct polynomials and the correctness of Cauchy's problem for dominantly correct partial differential operators with constant and variables coefficients in either two or more variables are given. Chapter 4 "Operators of principal type associated with Newton's polygon" deal with some generalizations and further development of the well-known results by Hörmander on polynomials and partial differential operators of principal type.

The second part (chapters 5-7) is devoted to Newton's polyhedron and its applications to the partial differential equations. Chapter 5 "Two-sided estimates in several variables relating to Newton's polyhedron" formally repeats chapter 1 and the part of chapter 2 but in multidimensional case. In particular the multidimensional N quasi-elliptic and N-parabolic polynomials and operators are studied. In analogs way chapter 6 "Operators of principal type associated with Newton's polyhedron" repeats for multidimensional case the main results of chapter 4. The last chapter 7 "The method of energy estimates in Cauchy's problem" is devoted to general analysis of the correctness property for Cauchy's problem of partial differential equations with variables coefficients (the results of foregone chapters dealt with only special and very stringent cases described by conditions of constant strength of the symbols of considered partial differential operators but these conditions are not fulfilled for strictly hyperbolic and dominantly correct differential operators with variable coefficients). The main part of chapter deal with the analysis of situations in which some energy estimates hold that are guaranteed the correctness of Cauchy's problem.

References in the book contains only 50 items; Index counts all special terms.

Undoubtedly in general the book is available for the beginning investigators and is of interested to specialists in field of partial differential or pseudodifferential operators and algebraic-analytical methods in different branches of mathematics.