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Partial differential equations: 1 Basic theory

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Bibliographic Details
Main Author:	Taylor, Michael Eugene 1946- (Author)
Format:	Book
Language:	English
Published:	New York [und andere] Springer [2010]
Edition:	Second edition
Series:	Applied mathematical sciences volume 115
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020537589&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Physical Description:	xxii, 654 Seiten
ISBN:	9781441970541 9781461427261

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adam_text	Contents Contents of Volumes II and III ................................................ xi Preface ............................................................................ xiii 1 Basic Theory of ODE and Vector Fields ................................. I 1 The derivative .......................................................... 3 2 Fundamental local existence theorem for ODK ....................... 9 3 Inverse function and implicit function theorems ...................... 12 4 Constant-coefficient linear systems; exponentiation of matrices __ 16 5 Variable-coefficient linear systems ofODl·: Duhameľs principle 26 6 Impendence of solutions on initial data and on other parameters З І 7 Rows and vector fields ................................ VS 8 Lie brackets ...................................................... 40 9 Commuting flows; Frobenius s theorem ................. 43 10 Hamiltonian systems ........................................... 47 I I Geodesies ............................................................ S I 12 Variational problems and the stationary· action principle ............ 59 13 Differential forms ...................................................... 70 14 The symplectic form and canonical transformations ................. 83 15 First-order, scalar, nonlinear PDE ..................................... 89 16 Completely integrable hamiltonian systems .......................... 96 17 Examples of integrable systems; central force problems ............. 101 18 Relativislic motion ..................................................... 105 19 Topologica] applications of differential forms ........................ 110 20 Critical points and index of a vector field ............................. 118 A Nonsmooth vector fields ............................................... 122 References .............................................................. 125 2 The Laplace Equation and Wave Equation ............................. 127 1 Vibrating strings and membranes ...................................... 129 2 The divergence of a vector field ....................................... 140 3 The cevariam derivative and divergence of tensor fields ............. 145 4 The Laplace operator on a Riemannian manifold .................... 153 5 The wave equation on a product manifold and energy conservation 156 6 Uniqueness and finite propagation speed ............................. 162 7 Lorentz manifolds and stress-energy tensors ......................... 166 8 More general hyperbolic equations; energy estimates ................ 172 Contents 9 The symbol of a differential operator and a genera] Green- Stokes formula ................................................. 176 10 The Hodge Laplacian on к -forms ..................................... 180 11 Maxwell s equations ................................................... 184 References .............................................................. 194 Fourier Analysis, Distributions, and Constant-Coefficient Linear PDE ................................... 197 1 Fourier series ........................................................... 198 2 Harmonic functions and bolomorphic functions in the plane ........209 3 The Fourier transform ..................................................222 4 Distributions and tempered distributions ..............................230 5 The classical evolution equations .....................................244 6 Radial distributions, polar coordinates, and Bessel functions ........263 7 The method of images and Poisson s summation formula ...........273 8 Homogeneous distributions and principal value distributions .......278 9 Elliptic operators .......................................................286 10 Local solvability of constant-coefficient PDE ........................289 11 The discrete Fourier transform ........................................292 12 The fast Fourier transform .............................................301 A The mighty Gaussian and the sublime gamma function ..............306 References ..............................................................312 Sobolev Spaces ..............................................................315 1 Sobolev spaces ont ..................................................315 2 The complex interpolation method ....................................321 3 Sobolev spaces on compact manifolds ................................328 4 Sobolev spaces on bounded domains .................................331 5 The Sobolev spaces H$ (Ω) ...........................................338 6 The Schwartz kernel theorem ..........................................345 7 Sobolev spaces on rough domains .....................................349 References ..............................................................351 Linear Elliptic Equations ..................................................353 1 Existence and regularity of solutions to the Dinchlet problem ......354 2 The weak ала strong maximum principles ............................364 3 The Dinchlet problem on the ball ml ..............................373 4 The Riemann mapping theorem (smooth boundary) .................379 5 The Dinchlet problem on a domam with a rough boundary .........383 6 The Riemann mapping theorem (rough boundary) ...................398 7 The Neumann boundary problem .....................................402 8 The Hodge decomposition and harmonic forms ......................410 9 Natural boundary problems for the Hodge Laplacian ................421 10 Isothermal coordinates and conformai structures on surfaces .......438 11 General elliptic boundary problems ...................................441 12 Operator properties of regular boundary problems ...................462 Contents їх A Spaces of generalized functions on manifolds with boundary .......471 В The Mayer-Vietoris sequence in deRham cohomoiogy ..............475 References ..............................................................478 6 Linear Evolution Equations ...............................................48 ] 1 The heat equation and the wave equation on bounded domains .....482 2 The heat equation and wave equation on unbounded domains ......490 3 Maxwell s equations ...................................................496 4 The Cauchy-Kowalewsky theorem ...................................499 5 Hyperbolic systems ....................................................504 6 Geometrical optics .....................................................510 7 The formation of caustics .............................................. 518 8 Boundary layer phenomena for the heat semigroup .................. 535 A Some Banach spaces of harmonic functions .......................... 541 В The stationary phase method .......................................... 543 References ..............................................................545 A Outline of Functional Analysis ............................................ 549 1 Banach spaces ..........................................................549 2 Hubert spaces .......................................................... 556 3 Fréchet spaces; locally convex spaces .................................561 4 Duality ..................................................................564 5 Linear operators ........................................................571 6 Compact operators .....................................................579 7 Fredholm operators ....................................................593 8 Unbounded operators ..................................................596 9 Semigroups .............................................................603 References ..............................................................615 В Manifolds, Vector Bundles, and Lie Groups .............................617 1 Metric spaces and topo logical spaces .................................617 2 Manifolds ...............................................................622 3 Vector bundles ..........................................................624 4 Sard s theorem ..........................................................626 5 Lie groups ..............................................................627 6 The Campbell-Hausdorff formula ....................................630 7 Representations of Lie groups and Lie algebras ......................632 8 Representations of compact Lie groups ...............................636 9 Representations of SU(2) and related groups .........................641 References ..............................................................647 Index ..............................................................................649
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spellingShingle	Taylor, Michael Eugene 1946- Partial differential equations Applied mathematical sciences
title	Partial differential equations
title_auth	Partial differential equations
title_exact_search	Partial differential equations
title_full	Partial differential equations 1 Basic theory Michael E. Taylor
title_fullStr	Partial differential equations 1 Basic theory Michael E. Taylor
title_full_unstemmed	Partial differential equations 1 Basic theory Michael E. Taylor
title_short	Partial differential equations
title_sort	partial differential equations basic theory
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