Mathematical Analysis and Numerical Methods for Science and Technology

Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo

Robert Dautray J acques-Louis Lions

Mathematical Analysis and Numerical Methods for Science and Technology Volume 1 Physical Origins and Classical Methods

With the Collaboration of Philippe Benilan, Michel Cessenat, Andre Gervat, Alain Kavenoky, Helene Lanchon

Translated from the French by Ian N. Sneddon

i Springer

Robert Dautray 12 rue du Capitaine Scott 75015 Paris, France

Jacques-Louis Lions College de France 3 rue d'Ulm 75231 Paris Cedex 5, France

Title of the French original edition: Analyse mathematique et calcul numerique pour les sciences et les techniques, Masson, S. A. © Commissariat it I 'Energie Atomique, Paris 1984, 1985

With 41 Figures

Mathematics Subject Classification (1980): 31-XX, 35-XX, 41-XX, 42-XX, 44-XX, 45-XX, 46-XX, 47-XX, 65-XX, 73-XX, 76-XX, 78-XX, 80-XX, 81-XX

Library of Congress Cataloging·in·Publication Data Dautray, Robert. Mathematical analysis and numerical methods for science and technology. Translation of: Analyse mathematique et calcul numerique pour les sciences et les techniques. Includes bibliographical references. Contents: v. 1. Physical Origins and Classical Methods - v. 2 Functional and varitional methodsl with the collaboration of Michel Artola ... let al.l I. Mathematical analysis. 2. Numerical analysis. 1. Lions. Jacques Louis. l. Title. QA300.D34313 1990 515 88-15089

ISBN-13: 978-3-540-66097-2 e-ISBN-13: 978-3-642-61527-6 DOl: 10.1007/978-3-642-61527-6

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifi­cally of translation, reprinting, reuse of illustrations, recitation, broadcasting. reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.

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SPIN: 10732803 41/3143-543 2 1 - Printed on acid-free paper

Preface

In the first years of the 1970's Robert Dautray engaged in conversations with Jacques Yvon, High-Commissioner of Atomic energy, of the necessity of publish­ing mathematical works of the highest level to put at the disposal of the scientific community a synthesis of the modern methods of calculating physical phe­nomena. It is necessary to get away from the habit of treating mathematical concepts as elegant abstract entities little used in practice. We must develop a technique, but without falling into an impoverishing utilitarianism. The competence of the Commissariat a I'Energie Atomique in this matter can provide a support of exceptional value for such an enterprise. The work which I have the pleasure to present realises the synthesis ofmathemat­ical methods, seen from the angle of their applications, and of use in designing computer programs. It should be seen as complete as possible for the present moment, with the present degree of development of each of the subjects. It is this specific approach which creates the richness of this work, at the same time a considerable achievement and a harbinger of the future. The encounter to which it gives rise among the originators of mathematical thought, the users of these concepts and computer scientists will be fruitful for the solution of the great problems which remain to be treated, should they arise from the mathematical structure itself (for example from non-linearities) or from the architecture of computers, such as parallel computers. This task has led to planning, spread over ten consecutive years of strenous work, by two exceptional men -- the physicist Robert Dautray and the mathematician Jacques-Louis Lions. In addition, they have enlisted the assistance of younger research workers, so it is fair to include them in our thanks for a work which, deemed indispensible thoughout, does not seem to me to have been undertaken quite at this level anywhere else in the world.

Jean Teillac High Commissioner of Atomic Energy

General Introduction

1. A very great number of the problems of mathematical physics can be "mod­elled" by partial differential equations. By a "model", we mean a set of equations (or inequations) which, together with boundary conditions (expressed on the boundary of the spatial domain where the phenomenon is studied) and, when the phenomenon is evolutionary, with initial conditions, allow us to define the state of the system. This is also called modelling by "distributed systems". Naturally the description of the model (or of a model, since the same phe­nomenon can often be described, in conditions not always strictly equivalent, by different state variables) is an important - but not decisive - step. Further, we must "study" the model, i.e. deduce qualitative or quantitative prop­erties which (a) recover, in simple conditions, observations (measurements) already made. (b) give supplentary information about the system. It has been observed for a long time that the majority of the phenomena of mathematical physics are non-linear, among the most celebrated cases being Boltzmann's equation in statistical mechanics, the Navier-Stokes in fluid mechan­ics (equations which moreover constitute an approximation to Boltzmann's equa­tion) von Karman's equations governing the large displacements of flat plates, etc. However, having the possibility of using in a systematic - and almost "common­place" way the procedures for calculating approximate solutions of the state of the system, precise results can generally only be obtained in [he linear cases. Certain physical problems can be modelled directly (i.e. without approximations) by linear equations: this is notably the case of the equation of transport of neutrons. Other phenomena can be deduced from "truly" non-linear systems by neglecting certain terms (which is valid in certain situations: "small" dis­placements, "slow" motions ... ) produced by linearisation about a particular solution. As, in addition, the methods brought into play for the solution of linear problems play an essential role in all the non-linear situations known to this day, it is indispensible to begin with the study of linear distributed models, or again with boundary value problems for linear partial differential equations (with brief in­cursions into the domain of linear integral equations, equations which we can deduce from linear partial differential equations, or can appear directly in this form). It is the aim of this work to study linear distributed models, completely concen­trating in particular on physical examples (from various sources), by the general

VIII General Introduction

methods ollinear analysis (stating very clearly the application of these methods to physically important situations), We have tried to render the material accessible to a reader the level of whose knowledge is pretty nearly that of the Lebesgue integral; the (indispensible) theory of distributions is recalled in the Appendix to Volume 2.

2. The theory of partial differential equations constitutes today one of the impor­tant topics of scientific understanding. The principal reasons for this state of affairs are, on the one hand the progress of mathematical analysis and, on the other hand, the arrival of the technique ol numerical calculus which remained, for partial differential equations, almost to­tally inadequate until the 1950's. In effect, the arrival of computers, and their immense and unceasing progress, have allowed us - for the first time in history

to calculate, beginning with the models, quantities which, formerly we were able only to estimate very approximately and, perhaps over all, to calculate them accurately and rapidly, and hence the (fundamental) possibility for r.esearch workers and engineers to be able to use the numerical results for the modification or adaptation of scientific arguments, of experiments or of constructions in progress. All that explains why, in very differing subjects, modelling by partial diflerentia! equations, followed by theoretical analysis, then numerical analysis, and then in its turn with comparison with experiment has become a basic method olprocedure. Every aspect of technical and industrial activity is concerned; this procedure is indispensible in the preparation or experiments and of trials and their interpreta­tion, technical studies, the development of manufacturing processes, mainte­nance, reliability, etc .. ,. Thus: Modern equipment has to operate in high performance with certain materials. In the 1950's, the calculation of the strength of materials was carried out with high safety factors (for example, 5 or more) on the stress experienced by the material of a given piece at a given point. Today, when we calculate a stress with precision, the safety factor which we take is of the order of 1.4 or less (for example in aeronautical, nuclear, automobile engineering etc .... ) and then in the very best conditions for users. Similarly, the reliability and security demanded by many modern techniques, from nuclear engineering to aeronautics, from aerospace to large public works (high speed rail transport, highway construction, generation and distribution of electricity, etc .... ) require the accumulation both of the safety factors and, as well require that each of the details is studied and is represented with great precision. No element is any longer "neglected", then and only then the faithful mathematical representation allows us to examine closely the least detail and underline the predictions. Modelling by distributed systems has become similarly the basis of many disci­plines in physics (plasmas, new materials, etc .... ) in the space and earth sciences (astrophysics, geophysics etc .... ), in chemistry and obviously in all branches of mechanics (a number of which have already been cited above). Without wishing to draw up here an exhaustive list, we should add that, by the intervention, notably of dynamic programming, (non-linear) partial differen-

General Introduction IX

tial equations play an important role in the manaf?ement sciences (stocks, energy, etc .... ). Distributed models are similarly involved, and more and more, in the lile sciences.

3. Plan of the Work. We give here a general sketch of the content of the chapters grouped by volumes. Each volume begins with a slightly more detailed account of its contents. We begin by giving in Chapter I a list of mathematical models, important in applications to physics and to the mechanics of continuous media which can lead to linear problems. The study of stationary linear problems begins with a review, in Chapter II, of the possibilities of making use of classical methods. We discover that their limits are quickly reached. We examine in Chapter III the possibilities of applying junc­tional tran~lormations (Fourier series, the Fourier, Mellin and Hankel transforms etc .... ). We touch there similarly on the limitations on their application. These limitations show the usefulness of working on sets (of distributions) very much more "extensive" than the sets of continuous functions considered in Chapter II: these are the spaces of general distributions introduced in Chapter III and the Sobolev spaces studied in Chapter IV. The study of diflerential operators in the spaces of general distributions allows us to distinguish the properties ofthese operators (elliptic, parabolic and hyperbolic operators; local character of mathematical models using differential operators; characteristics etc .... ) which will serve us well throughout this work; this is the subject matter of Chapter V. Throughout the whole of this work we shall have to handle operators; "opera­tions" on these operators and their approximations are explained, in Chapter VI, in the mathematical situations used in the present work. The mathematical techniques thus gathered together allow us to treat variational methods, which make up the subject of Chapter VII and whose potential for application extends to many non-linear problems. Numerous spectral problems arise in applications (calculation of energy levels and states in quantum mechanics, critical conditions in neutronics, transmission in a wave guide or in an optical fibre etc .... ). The spectral theory which enables us to treat such problems is seen in Chapter VHI within the perspective of typical applications; it includes especially the study of the continuous spectrum, source of many difficulties. Examples of applications are give in Chapter IX. A problem which is elliptic or hyperbolic according to the value of a parameter is treated in Chapter X: Tricomi's problem (in fluid mechanics it corresponds to the passage from subsonic flight to supersonic flight). Mathematical models involving integrals permit the representation of actions at a distance (in physical space, electric potential; in time, the memory of a viscoelastic body; in the space of velocities, change of velocity as a result of collisions). Integral equations which come into play require the methods treated in Chapter IX. Finally, the numerical methods to treat stationary problems form the subject matter of Chapters XlI and XIII.

x General I n trod uction

Linear evolution problems are treated first of all in the whole physical space in Chapter XIV. The diagonalisation method, using the spectral theory of operators, which is the basis of several practical methods (giving rise to the decomposition into modes), is treated in Chapter XV. The method of the Lap/ace trans!(Jrm can be used to treat numerous evolution problems; it is considered in Chapter XVI. The solution u (t) of a large class of evolution problems can be written in the form u (t) = G ({) U o ' where Uo is the initial value of u and Uo is the initial value of u and G (t) a family of operators forming a semi-group. The types of evolution of solutions can then be an examination of the various families of semi-groups G (t). This provides a method, in certain ways more general than the preceding ones, to treat evolution problems. This is the subject of Chapter XVII. Finally, the constructive methods of solving evolution problems (using construc­tions of solution in finite-dimensional spaces), the variational methods, are seen in Chapter XVIII 1. The Navin-Stokes problem (in the linearised case) requires particular variational methods. These are described in Chapter XIX. Chapter XX presents the numerical methods for linear evolution problems. The problems involving a transport equation are not included in the categories treated in Chapters XIV to XVIII, since they take into account the very particular type of properties of the transport operator (transport of neutrons, transport of molecules and Boltzmann's equation, transport of charged particles and Vlasov's equation). A special chapter, Chapter XXI. is therefore devoted to these prob­lems. Later chapters study other aspects of certain of the problems studied in the present work (relations between problems of partial differential equations and probabilities, propagation of waves, etc .... ).

4. The writing of this work has been conceived with the object of making it accessible to an engineer or to an aspiring research worker taking only the infor­mation he needs to treat his problem; a restricted reading is therefore possible if the reader is guided by the index, the table of contents and the table of notations.

5. In producing this work the undersigned have benifitted from the collaboration of many colleagues: Michel Artola, Marc Authier, Claude Bardos. Philippe Beni­lan, Michel Bernadou, Michel Cessenat. Jean-Michel Combes, Andre Gervat, Alain Kavenoky, Heh~ne Lanchon, Patrick Lascaux, Bertrand Mercier, Jean­Claude NedClec, Olivier Pironneau, Jacques Planchard, Bruno Scheurer, Claude Wild, Claude Zuily. Their contributions and the contributions on specific points due to several other colleagues will be acknowledged at the beginning of each volume. The manuscript was read with particular care by Michel Cessenat whom we thank most warmly. Considering the size and the diversity of this work. the task he performed is considerable. In addition, Michel Cessenat proposed complemen­tary or corrected texts, valuable contributions which we have often retained.

1 These methods can be similarly extended to non-linear problems.

General Introduction XI

Our thanks go similarly to Jean-Marie Moreau for his important work in compil­ing the bibliographies for each volume, for reading the text and bringing it to the point of publication. This work would not have seen the light of day without the support of the Atomic Energy Commission (C.E.A.): Jacques Yvon, then High Commissioner of Atomic Energy accepted our proposition immediately, as he could foresee its future development. He made its publication one of the scientific enterprises of the C .E.A. Our respective experiences had, in effect, as early as the end of the 1960's, confirmed our belief in the importance of the existence of a work of reference of this type. By the beginning of 1970's, we had elaborated our ideas into a plan, taking account of the needs of engineers, physicists and workers in mechanics etc .... Jacques Yvon together with ourselves, wished to spread and put within their grasp the abundant recent work of mathematicians and numerical analysts. In the initial period, at the time of preliminary drafts and launching the project, we benifitted from the initiative of Robert Lattes, who was then Scientific Adviser of the C.E.A. We are grateful to Paul Bonnet, Inspector General of the C.E.A. for having inaugurated the C.E.A. collection with this work. We have greatly valued, and are immensely grateful for, the initial help and encouragement of Jules Horowitz, Director at the C.E.A. who with his great experience in mathematical physics showed an immediate understanding of our alms. Nothing would have been achieved in reaching the final result without the clear and active understanding of Jacques Chevallier, Director at the C.E.A. We thank here also Michel Pecquer and Gerard Renon, Administrator General, as well as Jean Teillac, High Commissioner, of the C.E.A. whose constant and manifest approval, personally expressed, has been a source of permanent encour­agement.

R. Dautray, J.-L. Lions

Practical Guide for the Reader

(1) Designation of the subdivisions of the text: Number of chapters: in roman numerals; N umber of major divisions: the sign § followed by a numeral; Number of sections: a numeral following the preceding; Number of sub-sections: a numeral following the preceding; etc .... For example: II, § 3.5.2, denotes Chapter II, § 3, section 5, subsection 2. (2) In the interior ()leach division (§), the equations, definitions, theorems, propo­sitions, corollaries, lemmas, remarks and examples are numbered separately in sequence beginning with the numeral 1. (3) The table of {he notations used appears at the end of each volume.

Introduction to Volume 1

Chapter I gives the principal physical examples studied in this work (these exam­ples come from physics, from mechanics, from chemistry, etc .... ). A first (rudi­mentary) attempt at the classification of the problems is made. In all the phenomena modelled by partial differential equations, and for reasons that are given in the text, a very important role is played by the Laplacian operator

a2 a2 a2

L1=-+-~+-axi ax~ ax~

in rectangular coordinates: This is why Chapter II is devoted to a direct study of the principal questions linked with this operator. "direct" signifying here: without the use of techniques other than those of classical analysis. We give below the authors of various contributions to these two chapters.

Chapter I: H. Lanchon, M. Cessenat, A. Gervat, A. Kavenoky. Chapter II: P. Benilan, sole author of this Chapter.

We similarly wish to thank R. Balian, C. Bardos, A. Bossavit, C. Cohen-Tan­noudji, G. Fournet, A. Kavenoky and E. Roubine for reading certain portions of the text and for their advice on modifying them.

The reader wishing to acquaint himself rapidly with the essential mathematical and numerical methods should be able to make use of this volume and the subsequent Vol. 2 by leaving for a later, deeper study §§ 5-8 of Chapter II of this volume and §§ 4, 5 of Chapter V of Vol. 2. These divisions are denoted by an asterisk * placed at their beginning, an asterisk which, moreover, appears in the table of contents. We have placed the table of notations at the end of this volume.

R. Dautray, J.-L. Lions

Table of Contents

Chapter I. Physical Examples

Introduction . . . . . . .

Part A. The Physical Models 2

§1. Classical Fluids and the Navier-Stokes System 2 1. Introduction: Mechanical Origin . . . 2 2. Corresponding Mathematical Problem. . 5 3. Linearisation. Stokes' Equations . . . . 7 4. Case of a Perfect Fluid. Euler's Equations 7 5. Case of Stationary Flows. Examples of Linear Problems 9 6. Non-Stationary Flows Leading to the Equations of Viscous

Diffusion . . . . . . . . . . . . . . . . . . . . . . 15 7. Conduction of Heat. Linear Example in the Mechanics of Fluids 18 8. Example of Acoustic Propagation. . . . . . . . . . . . 23 9. Example with Boundary Conditions on Oblique Derivatives 25 Review . . . . 27

§2. Linear Elasticity 28

1. Introduction: Elasticity; Hyperelasticity 28 2. Linear (not Necessarily Isotropic) Elasticity 29 3. Isotropic Linear Elasticity (or Classical Elasticity) 32 4. Stationary Problems in Classical Elasticity . . . 33 5. Dynamical Problems in Classical Elasticity. . . 37 6. Problems of Thermal Diffusion. Classical Thermoelasticity 47 Review . . . . . . 51

§3. Linear Viscoelasticity 52 1. Introduction . . . 52 2. Materials with Short Memory 53 3. Materials with Long Memory 53 4. Particular Case of Isotropic Media 54 5. Stationary Problems in Classical Viscoelasticity 59 Review . . . . . . . . . . . . . . . . 59

§4. Electromagnetism and Maxwell's Equations 59 1. Fundamental Equations of Electromagnetism 59 2. Macroscopic Equations: Electromagnetism in Continuous Media 65

XIV Table oj' Contents

3. Potentials. Gauge Transformation (Case of the Entire Space lR; x lR,) 80

4. Some Evolution Problems 83 5. Static Electromagnetism 87 6. Stationary Problems 91 Review 111

§5. Neutronics. Equations of Transport and Diffusion 113

1. Problems of the Transport of Neutrons 113 2. Problems of Neutron Diffusion 117 3. Stationary Problems 121 Review 126

§6. Quantum Physics 128

Introduction . . 128

1. The Fundamental Principles of Modelling 130 2. Systems Consisting of One Particle 142 3. Systems of Several Particles 152 Review . . . . 157 Appendix. Concise Elements Concerning Some Mathematical Ideas Used in this §6 . . . . . . . . . . . . . . . . . . . . . .. 157

Appendix "A1echanics'". Elements Concerning the Prohfcl1ls of" Mcchanics. . . . .. ............ 162

§ 1. Indicia I Calculus. Elementary Techniques of the Tensor Calculus. 162

1. Orientation Tensor or Fund.amental Alternating Tensor in lR 3 162 2. Possibilities of Decompositions of a Second Order Tensor 164 3. Generalized Divergence Theorem 165 4. Ideas About Wrenches .... 166

§2. Notation, Language and Conventions in Mechanics 167

1. Lagrangian and Eulerian Coordinates . . 167 2. Notions of Displacement and of Strain 168 3. Notions of Velocity and of Rate of Strain 169 4. Notions of Particle Derivative, of AcceleratiQn and of Dilatation 170 5. Notions of Trajectory and of Stream Line . . 171

§3. Ideas Concerning the Principle of Virtual Power. 172

1. Introduction: Schematization of Forces 172 2. Preliminary Definitions 172 3. Fundamental Statements ... . . . 175 4. Theory of the First Gradient . . . . . 176 5. Application to the Formulation of Curvilinear Media 179 6. Application to the Formulation of the Theory of Thin Plates 183

Linear and Non-Linear Problems in §1 to §6 of this Chapter IA. . 187

Table of Contents xv

Part B. First Examination of the Mathematical Models 191

§ 1. The Principal Types of Linear Partial Differential Equations Seen in Chapter IA . . . . . . . . . . . . . 191

1. Equation of Diffusion Type . . . . . . 192 2. Equation of the Type of Wave Equations 195 3. Schr6dinger Equation. . . . . . . . . 197 4. The Equation Au = f in which A is a Linear Operator not

Depending on the Time and f is Given (Stationary Equations) 198

§2. Global Constraints Imposed on the Solutions of a Problem: Inclu-sion in a Function Space; Boundary Conditions; Initial Conditions. 201

1. Introduction. Function Spaces . . . . . 201 2. Initial Conditions and Evolution Problems 202 3. Boundary Conditions . . . . . . . . . 204 4. Transmission Conditions ....... 213 5. Problems Involving Time-Derivatives of the Unknown Function u

on the Boundary . . . . 215 6. Problems of Time Delay. 216

Review of Chapter IB . . . . . 218

Chapter II. The Laplace Operator

Introduction . . . . . . 220

§ 1. The Laplace Operator 220

1. Poisson's Equation 220 2. Examples in Mechanics and Electrostatics 224 3. Green's Formulae: The Classical Framework 226 4. The Laplacian in Polar Coordinates. . . . 231

§2. Harmonic Functions ........... 236

1. Definitions. Examples. Elementary Solutions 236 2. Gauss' Theorem. Formulae of the Mean. The Maximum

Principle . . . . . . . . . . . . . . . . . . . . 243 3. Poisson's Integral Formula; Regularity of Harmonic Functions;

Harnack's Inequality . . . . . . . . . . . . . . . . 248 4. Characterisation of Harmonic Functions. Elimination of

Singularities . . . . . . . . . . . . . . . . . . . . 255 5. Kelvin's Transformation; Application to Harmonic Functions

in an Unbounded Set; Conformal Transformation. . . . .. 264 6. Some Physical Interpretations (in Mechanics and Electrostatics). 271

§3. Newtonian Potentials . . . . . . . . . . . . . . . . . . .. 275 1. Generalities on the Newtonian Potentials of a Distribution with

Compact Support .. . . . . . . . . . . . . . . . .. 275 2. Study of Local Regularity of Solutions of Poisson's Equation.. 283

XVI Table of Contents

3. Regularity of Simple and Double Layer Potentials ..... 296 4. Newtonian Potential of a Distribution Without Compact Support 307 5. Some Physical Interpretations (in Mechanics and Electrostatics) 326

§4. Classical Theory of Dirichlet's Problem . . . . . . . . . . 328

1. Generalities on Dirichlet's Problem P(Q, cp) in the Case Q

Bounded: Classical Solution, Examples, Outline of Perron's Method, Generalized Solutions, Regular Point of the Boundary. Barrier Function . . . . . . . . . . . . . . . . . . . .. 328

2. Generalities on the Dirichlet Problem P(Q, cp, f) and the Green's Function of Q, a Bounded Open Set ...... . . . .. 341

3. Generalities on Dirichlet's Problem in an Unbounded Open Set 352 4. The Neumann Problem; Mixed Problem; Hopf's Maximum

Principle; Examples. . . . . . . . . . . . . . . . . . 367 5. Solution by Simple and Double Layer Potentials: Fredholm's

Integral Method . . . . . . . . . . . . 379 6. Sub-Harmonic Functions. Perron's Method 397

§5. Capacities*. . . . . . . . . . . . . . . . 407 1. Interior and Exterior Capacity Operators 407 2. Electrical Equilibrium; Coefficients of Capacitance 418 3. Capacity of a Part of an Open Set in IR" 433

§6. Regularity"'. . . . . . . . . . . . . . 457

1. Regularity of the Solutions of Dirichlet and Neumann Problems 457 2. Analytic Regularity and Trace on the Boundary of a Harmonic

Function . . . . . . . . . . . . . . . . . . . . . 472 3. Dirichlet Problem with Given Measures or Discontinuous

Functions. Herglotz's Theorem . . . . . . . . . . . . 483 4. Neumann Problem with Given Measures ....... 497 5. Dependence of Solutions of Dirichlet Problems as a Function

of the Open Set: Hadamard's Formula . . . . . 502

§7. Other Methods of Solution of the Dirichlet Problem*. 507

1. Case of a Convex Open Set: Neumann's Integral Method 507 2. Alternating Procedure of Schwarz .......... 515 3. Method of Separation of Variables. Harmonic Polynomials.

Spherical Harmonic Function . . . . . . 523 4. Dirichlet's Method . . . . . . . . . . . 542 5. Symmetry Methods and Method of Images 560

§8. Elliptic Equations of the Second Order* . . . 568 I. The Divergence Form, Green's Formula. . 570 2. Different Concepts of Solutions, Boundary Value Problems,

Transmission Conditions ........... . . . . 576 3. General Results on the Regularity of Elliptic Problems of the

Second Order . . . . . . . . . . . . . . . . . . . . 585

Table of Contents XVII

4. Results on Existence and Uniqueness of Solutions of Strictly Elliptic Boundary Value Problems of the Second Order on a Bounded Open Set . . . . . . . . . . . . . 594

5. Harnack's Inequality and the Maximum Principle 607 6. Green's Functions 624 7. Helmholtz's Equation 640

Review of Chapter II 658

Bibliography. . . 659

Table of Notations 667

Index . . . . . . 681

Contents of Volumes 2-6 715