Partial Differential Equations 2: Functional Analytic MethodsThis comprehensive two-volume textbook presents the whole area of Partial Differential Equations - of the elliptic, parabolic, and hyperbolic type - in two and several variables. Special emphasis is put on the connection of PDEs and complex variable methods. In this second volume the following topics are treated: Solvability of operator equations in Banach spaces, Linear operators in Hilbert spaces and spectral theory, Schauder's theory of linear elliptic differential equations, Weak solutions of differential equations, Nonlinear partial differential equations and characteristics, Nonlinear elliptic systems with differential-geometric applications. While partial differential equations are solved via integral representations in the preceding volume, functional analytic methods are used in this volume. This textbook can be chosen for a course over several semesters on a medium level. Advanced readers may study each chapter independently from the others. |
Contents
| 1 | |
| 2 | |
| 3 | |
| 4 | |
| 5 | |
| 6 | |
| 7 | |
| 8 | |
2 Twodimensional parametric integrals | 265 |
3 Quasilinear hyperbolic differential equations and systems of second order Characteristic parameters | 274 |
Brouwers Degree of Mapping with Geometric Applica tions | 279 |
1 The winding number 2 The degree of mapping in Rn 3 Geometric existence theorems 4 The index of a mapping 5 The product theorem 6 Theore... | 280 |
4 Cauchys initial value problem for quasilinear hyperbolic differential equations and systems of second order | 281 |
5 Riemanns integration method | 291 |
6 Bernsteins analyticity theorem | 296 |
7 Some historical notices to chapter XI | 302 |
Linear Operators in Hilbert Spaces | 31 |
9 Some historical notices to chapter I | 125 |
Linear Elliptic Differential Equations | 127 |
Weak Solutions of Elliptic Differential Equations | 187 |
2 Embedding and compactness | 201 |
3 Existence of weak solutions | 208 |
4 Boundedness of weak solutions | 213 |
5 Hölder continuity of weak solutions | 216 |
6 Weak potentialtheoretic estimates | 227 |
7 Boundary behavior of weak solutions | 234 |
8 Equations in divergence form | 239 |
9 Greens function for elliptic operators | 245 |
10 Spectral theory of the LaplaceBeltrami operator | 254 |
11 Some historical notices to chapter X | 256 |
Nonlinear Partial Differential Equations 259 | 258 |
Nonlinear Elliptic Systems | 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 Plateaus problem | 374 |
11 Some historical notices to chapter XII | 379 |
| 382 | |
| 385 | |
Other editions - View all
Partial Differential Equations: With Consideration of Lectures by E. Heinz ... Friedrich Sauvigny No preview available - 1975 |