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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