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The Method of Newton's Polyhedron in the Theory of Partial Differential Equations : Mathematics and its Applications - Semen G. Gindikin

The Method of Newton's Polyhedron in the Theory of Partial Differential Equations

Mathematics and its Applications

Hardcover

Published: 30th November 1992
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One service mathematics has rendered the 'Et moi, .. ., si j'avait su comment cn rcvenir, human race. It has put common sense back. je n'y serais point aile.' where it bdongs, on the topmost shelf neAt Jules Verne to the dusty canister labelled 'discarded non· sense'. The series is divergent; therefore we may be Eric T. Bdl able to do something with it. O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non­ linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com­ puter science .. :; 'One service category theory has rendered mathematics .. .'. All a,rguably true. And all statements obtainable this way form part of the raison d'etre of this series.

Preface
Two-sided estimates for polynomials related to Newton's polygon and their application to studying local properties of partial differential operators in two variablesp. 1
Newton's polygon of a polynomial in two variablesp. 2
Polynomials admitting of two-sided estimatesp. 13
N Quasi-elliptic polynomials in two variablesp. 22
N Quasi-elliptic differential operatorsp. 27
Parabolic operators associated with Newton's polygonp. 49
Polynomials correct in Petrovskii's sensep. 50
Two-sided estimates for polynomials in two variables satisfying Petrovskii's condition. N-parabolic polynomialsp. 57
Cauchy's problem for N-stable correct and N-parabolic differential operators in the case of one spatial variablep. 64
Stable-correct and parabolic polynomials in several variablesp. 75
Cauchy's problem for stable-correct differential operators with variable coefficientsp. 81
Dominantly correct operatorsp. 93
Strictly hyperbolic operatorsp. 94
Dominantly correct polynomials in two variablesp. 97
Dominantly correct differential operators with variable coefficients (the case of two variables)p. 104
Dominantly correct polynomials and the corresponding differential operators (the case of several spatial variables)p. 111
Operators of principal type associated with Newton's polygonp. 118
Introduction. Operators of principal and quasi-principal typep. 118
Polynomials of N-principal typep. 124
The main L[subscript 2] estimate for operators of N-principal typep. 135
Local solvability of differential operators of N-principal typep. 150
Two-sided estimates in several variables relating to Newton's polyhedrap. 158
Estimates for polynomials in R[superscript n] relating to Newton's polyhedrap. 158
Two-sided estimates in some regions in R[superscript n] relating to Newton's polyhedron. Special classes of polynomials and differential operators in several variablesp. 169
Operators of principal type associated with Newton's polyhedronp. 177
Polynomials of N-principal typep. 177
Estimates for polynomials of N-principal type in regions of special formp. 186
The covering of R[superscript n] by special regions associated with Newton's polyhedronp. 193
Differential operators of N-principal type with variable coefficientsp. 200
The method of energy estimates in Cauchy's problemp. 215
Introduction. The functional scheme of the proof of the solvability of Cauchy's problemp. 215
Sufficient conditions for the existence of energy estimatesp. 223
An analysis of conditions for the existence of energy estimatesp. 235
Cauchy's problem for dominantly correct differential operatorsp. 247
Referencesp. 261
Indexp. 265
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780792320371
ISBN-10: 0792320379
Series: Mathematics and its Applications
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 266
Published: 30th November 1992
Publisher: Springer
Country of Publication: NL
Dimensions (cm): 23.5 x 15.5  x 1.91
Weight (kg): 0.58