Ⅶ Operators in Banach Spaces 1
1 Fixed point theorems 1
2 The Leray-Schauder degree of mapping 12
3 Fundamental properties for the degree of mapping 18
4 Linear operators in Banach spaces 22
5 Some historical notices to the chapters Ⅲ and Ⅶ 29
Ⅷ Linear Operators in Hilbert Spaces 31
1 Various eigenvalue problems 31
2 Singular integral equations 45
3 The abstract Hilbert space 54
4 Bounded linear operators in Hilbert spaces 64
5 Unitary operators 75
6 Completely continuous operators in Hilbert spaces 87
7 Spectral theory for completely continuous Hermitian operators 103
8 The Sturm-Liouville eigenvalue problem 110
9 Weyl's eigenvalue problem for the Laplace operator 117
9 Some historical notices to chapter Ⅷ 125
Ⅸ Linear Elliptic Differential Equations 127
1 The differential equation ⊿φ＋p（x,y）φx＋q（x,y）φy＝r（x,y） 127
2 The Schwarzian integral formula 133
3 The Riemann-Hilbert boundary value problem 136
4 Potential-theoretic estimates 144
5 Schauder's continuity method 156
6 Existence and regularity theorems 161
7 The Schauder estimates 169
8 Some historical notices to chapter Ⅸ 185
Ⅹ Weak Solutions of Elliptic Differential Equations 187
1 Sobolev spaces 187
2 Embedding and compactness 201
3 Existence of weak solutions 208
4 Boundedness of weak solutions 213
5 H？lder continuity of weak solutions 216
6 Weak potential-theoretic estimates 227
7 Boundary behavior of weak solutions 234
8 Equations in divergence form 239
9 Green's function for elliptic operators 245
10 Spectral theory of the Laplace-Beltrami operator 254
11 Some historical notices to chapter Ⅹ 256
Ⅺ Nonlinear Partial Differential Equations 259
1 The fundamental forms and curvatures of a surface 259
2 Two-dimensional parametric integrals 265
3 Quasilinear hyperbolic differential equations and systems of second order （Characteristic parameters） 274
4 Cauchy's initial value problem for quasilinear hyperbolic differential equations and systems of second order 281
5 Riemann's integration method 291
6 Bernstein's analyticity theorem 296
7 Some historical notices to chapter Ⅺ 302
Ⅻ Nonlinear Elliptic Systems 305
1 Maximum principles for the H-surface system 305
2 Gradient estimates for nonlinear elliptic systems 312
3 Global estimates for nonlinear systems 324
4 The Dirichlet problem for nonlinear elliptic systems 328
5 Distortion estimates for plane elliptic systems 336
6 A curvature estimate for minimal surfaces 344
7 Global estimates for conformal mappings with respect to Riemannian metrics 348
8 Introduction of conformal parameters into a Riemannian metric 357
9 The uniformization method for quasilinear elliptic differential equations and the Dirichlet problem 362
10 An outlook on Plateau's problem 374
11 Some historical notices to chapter Ⅻ 379
References 383
Index 385