Variational Methods Boundary Value Problems Elliptic Equations

Description: Variational Methods for Boundary Value Problems for Systems of Elliptic Equations by M.A. Lavrent'ev P. Noordhoff Ltd Groningen, The Netherlands, 1963, Hardcover without Dust Jacket, Good condition, previous owner's name and info front endpaper, no underlining, no highlighting, 150 pages. FOREWORD At the present time, the problems of the existence and stability of the solutions of different classes of boundary value problems of the equations of mathematical physics continue to be of great interest. In particular, great progress has occurred during the past ten years in the domain of linear problems, where the method of integral equations with Fredholm's well-known alternative made it possible to arrive at final solutions of all basic linear problems for equations of the elliptic type; this method also greatly assisted the advance of the study of Tricomi's problem for equations of the mixed type. Beginning with the well-known studies of Villat, Levi-Civita and Nekrasov, we have a large group of investigations of classical non-linear problems of the mechanics of continuous media, of the problems of the flow around an arbitrary contour and of the wave motion of a heavy fluid. The majority of studies in this direction also employ non-linear integral equations with an application of the method of expansion with respect to a small parameter (Nekrasov, Kochin, etc.) or with an application of the methods of functional analysis, in particular, of the well-known fixed point theorem (Leray, Weinstein, Kravtchenko, etc.). In this monograph, the author presents an approach to these problems which is different in principle. It is based on a number of geometric properties of conformal and quasi-conformal mappings and uses a general basic scheme for the solution of variational problems, first suggested by Hilbert and developed by Tonnelli. The method lies on the boundary between the classical methods of analysis with its concrete estimates and approximate formulae and the methods of the theory of functions of a real variable with their special character and general theoretical quantitative aspects. On the basis of these facts, I wish to emphasize beforehand, that in the writing of the monograph I have tended to organize it in such a way that it may be used by mathematicians as well as by theoreticians in mechanics, for removed from function theory. Probably, the method of the proof of the existence and uniqueness theorems will be of interest to mathematicians (Chapters I–III) as well as the general theory of quasi-conformal mappings; mathematicians may readily omit those sections which refer to approximate methods and studies of particular problems in hydrodynamics and wave theory. Theoreticians in mechanics, far removed from function theory, can freely overlook the general scheme of the proof of the existence and uniqueness theorems and concentrate on the approximate formulae for conformal and quasi-conformal mappings which may turn otu to be useful (if combined with the methods for estimating errors proposed in the monograph) for the solution of many concrete problems of the mechanics of continuous media. I have not attempted to present the details of proofs and frequently have only given an outline of the general ideas. Likewise I have not aimed at the widest possible assumptions nor have I dwelt on the functional theoretical aspects of the problems. In conclusion I wish to thank Professor B. V. Shabat for his editorial help and for his highly valued comments relating to this book. Novosibirsk, 1960 M. A. LAVRENT'EV CONTENTS INTRODUCTION 1. Variational principles 2. Sufficient conditions 3. Generalizations Chapter I. VARIATIONAL PRINCIPLES OF THE THEORY OF CONFORMAL MAPPING 1.1. The principles of Lindelof and Montel 1.1.1. The case of the circle 1.1.2. Mapping on to a strip 1.2. Mechanical interpretation 1.3. Quantitative estimates Chapter II. BEHAVIOUR OF A CONFORMAL TRANSFORMATION ON THE BOUNDARY 2.1. Derivatives at the boundary 2.2. Narrow strips 2.3. Behaviour of the extension at points of maximum inclination and extreme curvature Chapter III. HYDRODYNAMIC APPLICATIONS 3.1. Stream line flow 3.2. Generalizations 3.3. Stream line flow with detachment 3.4. Wave motions of a fluid 3.5. The linear theory of waves 3.6. Rayleigh waves 3.7. The exact theory 3.8. Generalizations 3.8.1. Motion of a fluid over a submarine trench 3.8.2. Motion over a bottom with a ridge 3.8.3. Spillway with singularities Chapter IV. QUASI-CONFORMAL MAPPINGS 4.1. The concept of the quasi-conformal map 4.2. Derivative systems 4.3. Strong ellipticity Chapter V. LINEAR SYSTEMS 5.1. Transformations with bounded distortion 5.1.1. Equi-graded continuity 5.1.2. Almost conformal mappings . 5.2. The simplest class of linear systems 5.2.1. Invariance with respect to conformal mappings 5.2.2. Stability of conformal mappings 5.2.3. Condition of smoothness of a transformation 5.2.4. Application to arbitrary linear systems 5.2.5. Existence theorem Chapter VI. THE SIMPLEST CLASSES OF NON-LINEAR SYSTEMS 6.1. Maximum principle 6.2. The principle of Schwarz-Lindelof 6.3. Quantitative estimates 6.4. Inductive proof of Lindelof's principle 6.5. The existence theorem 6.6. Generalizations 6.7. Hydrodynamic applications REFERENCES

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