The theory of equations of mixed type originated in the fundamental researches of the Italian mathematician Francesco Tricomi, which were published in the twenties of this century. Owing to the importance of its applications, the discussion of problems concerned with equations of mixed type has become, in the last ten years, one of the central problems in the theory of partial differential equations.

The present work is not meant to be a summary of all results in this field, especially since the number of results increases with great speed; nevertheless, the reader of this monograph will obtain an idea of the present state of the theory of equations of mixed type.

This book was developed from a series of lectures dealing with certain fundamental questions in the theory of equations of mixed type, which the Author delivered in scientific establishments in the Chinese People’s Republic at the end of 1957 and the beginning of 1958.

Translated by P. Zador

Translation edited by I. N. Sneddon

You can get the book here and here.

CONTENTS

Foreword ix

Introduction xi

1. General remarks on linear partial differential equations of mixed type 1

1. Equation of the second order with two independent variables 1

2. The theory of Cibrario 3

3. Systems of two first order equations 12

4. Linear systems of partial differential equations of the second order with two independent variables 17

2. The study of the solutions of second order hyperbolic equations with initial conditions given along the lines of parabolicity 20

1. The Riemann function for a second order hyperbolic linear equation 20

2. A class of hyperbolic systems of second order linear equations 27

3. The Cauchy problem for hyperbolic equations with given initial conditions on the line of parabolic degeneracy 32

4. Generalizations 41

3. The study of the solutions of second order elliptic equations for a domain, the boundary of which includes a segment of the curve of parabolic degeneracy 44

1. The linear elliptic partial differential equation of the second order 44

2. Elliptic systems of second order 49

3. The Dirichlet problem for second order elliptic equations in a domain, the boundary of which includes a segment of the curve of parabolic degeneracy 58

4. Some other problems and generalizations 66

4. The problem of Tricomi 71

1. The statement of the problem of Tricomi 72

2. The extremal principle and the uniqueness of the solution of problem T 74

3. Solution of problem T by means of the method of integral equations 78

4. Continuation. The proof for the existence of a solution of the integral equations obtained in the preceding paragraph 90

5. Other methods for solving problem T 94

6. Examples and generalizations 103

5. Other mixed problems 112

1. The mixed problem M 112

2. The proof of the uniqueness of solution for problem M 113

3. Concerning the existence of the solution of problem M 117

4. The general mixed problem 124

5. The problem of Frankl 135

6. Short indication of some important generalizations and applications 141

References 151

Index 157