amann h , escher j analysis ii (birkhauser, 2008)(isbn 3764374721)(409s) mcetsinhvienzone com

om e C Zo n nh Vi en Si nhVienZone.com https://fb.com/sinhvienzonevn Zo ne Analysis II C om Herbert Amann Joachim Escher en Translated from the German Si nh Vi by Silvio Levy and Matthew Cargo Birkhäuser Basel · Boston · Berlin SinhVienZone.com https://fb.com/sinhvienzonevn Authors: Joachim Escher Institut für Angewandte Mathematik Universität Hannover Welfengarten 30167 Hannover Germany e-mail: escher@ifam.uni-hannover.de C om Herbert Amann Institut für Mathematik Universität Zürich Winterthurerstr 190 8057 Zürich Switzerland e-mail: herbert.amann@math.uzh.ch ne Originally published in German under the same title by Birkhäuser Verlag, Switzerland © 1999 by Birkhäuser Verlag Zo 2000 Mathematics Subject Classification: 26-01, 26A42, 26Bxx, 30-01 en Library of Congress Control Number: 2008926303 Vi Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at nh ISBN 3-7643-7472-3 Birkhäuser Verlag, Basel – Boston – Berlin Si This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks For any kind of use permission of the copyright owner must be obtained © 2008 Birkhäuser Verlag AG Basel · Boston · Berlin P.O Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced of chlorine-free pulp TCF d Printed in Germany ISBN 978-3-7643-7472-3 987654321 SinhVienZone.com e-ISBN 978-3-7643-7478-5 www.birkhauser.ch https://fb.com/sinhvienzonevn .C om Foreword en Zo ne As with the first, the second volume contains substantially more material than can be covered in a one-semester course Such courses may omit many beautiful and well-grounded applications which connect broadly to many areas of mathematics We of course hope that students will pursue this material independently; teachers may find it useful for undergraduate seminars For an overview of the material presented, consult the table of contents and the chapter introductions As before, we stress that doing the numerous exercises is indispensable for understanding the subject matter, and they also round out and amplify the main text Si nh Vi In writing this volume, we are indebted to the help of many We especially thank our friends and colleages Pavol Quittner and Gieri Simonett They have not only meticulously reviewed the entire manuscript and assisted in weeding out errors but also, through their valuable suggestions for improvement, contributed essentially to the final version We also extend great thanks to our sta2 for their careful perusal of the entire manuscript and for tracking errata and inaccuracies Our most heartfelt thank extends again to our “typesetting perfectionist”, without whose tireless e2ort this book would not look nearly so nice We also thank Andreas for helping resolve hardware and software problems Finally, we extend thanks to Thomas Hintermann and to Birkhă auser for the good working relationship and their understanding of our desired deadlines Ză urich and Kassel, March 1999 The H Amann and J Escher text was set in LATEX, and the figures were created with CorelDRAW! and Maple SinhVienZone.com https://fb.com/sinhvienzonevn vi Foreword Foreword to the second edition C om In this version, we have corrected errors, resolved imprecisions, and simplified several proofs These areas for improvement were brought to our attention by readers To them and to our colleagues H Crauel, A Ilchmann and G Prokert, we extend heartfelt thanks Ză urich and Hannover, December 2003 H Amann and J Escher ne Foreword to the English translation en Zo It is a pleasure to express our gratitude to Silvio Levy and Matt Cargo for their careful and insightful translation of the original German text into English Their effective and pleasant cooperation during the process of translation is highly appreciated H Amann and J Escher Si nh Vi Ză urich and Hannover, March 2008 SinhVienZone.com https://fb.com/sinhvienzonevn .C om Contents Chapter VI Jump continuous functions Staircase and jump continuous functions A characterization of jump continuous functions The Banach space of jump continuous functions en The extension of uniformly continuous functions Bounded linear operators The continuous extension of bounded linear operators 10 12 15 The Cauchy–Riemann Integral 17 The integral of staircase functions The integral of jump continuous functions Riemann sums 17 19 20 Properties of integrals 25 Integration of sequences of functions The oriented integral Positivity and monotony of integrals Componentwise integration The first fundamental theorem of calculus The indefinite integral The mean value theorem for integrals 25 26 27 30 30 32 33 The technique of integration 38 Variable substitution Integration by parts The integrals of rational functions 38 40 43 Vi 10 nh Continuous extensions Si v Integral calculus in one variable Zo ne Foreword SinhVienZone.com https://fb.com/sinhvienzonevn viii 50 50 52 53 54 56 57 59 64 Fourier series 67 The L2 scalar product Approximating in the quadratic mean Orthonormal systems Integrating periodic functions Fourier coe3cients Classical Fourier series Bessel’s inequality Complete orthonormal systems Piecewise continuously di2erentiable functions Uniform convergence 67 69 71 72 73 74 77 79 82 83 en Zo ne Sums and integrals The Bernoulli numbers Recursion formulas The Bernoulli polynomials The Euler–Maclaurin sum formula Power sums Asymptotic equivalence The Riemann function The trapezoid rule C om Contents Improper integrals Admissible functions Improper integrals The integral comparison test for Absolutely convergent integrals The majorant criterion series 90 90 90 93 94 95 The gamma function Euler’s integral representation The gamma function on C\(−N) Gauss’s representation formula The reflection formula The logarithmic convexity of the gamma function Stirling’s formula The Euler beta integral 98 98 99 100 104 105 108 110 Si nh Vi SinhVienZone.com https://fb.com/sinhvienzonevn Contents ix Chapter VII Multivariable differential calculus Continuous linear maps 118 Vi en 118 119 122 125 128 129 131 133 136 140 149 150 152 153 155 156 158 159 162 nh Si 166 166 169 169 171 171 Multilinear maps 173 Continuous multilinear maps The canonical isomorphism Symmetric multilinear maps The derivative of multilinear maps Multivariable di2erentiation rules 166 Linearity The chain rule The product rule The mean value theorem The differentiability of limits of sequences of functions Necessary condition for local extrema Differentiability 149 The definition The derivative Directional derivatives Partial derivatives The Jacobi matrix A differentiability criterion The Riesz representation theorem The gradient Complex differentiability Zo C om The completeness of L(E, F ) Finite-dimensional Banach spaces Matrix representations The exponential map Linear di2erential equations Gronwall’s lemma The variation of constants formula Determinants and eigenvalues Fundamental matrices Second order linear di2erential equations ne 173 175 176 177 Higher derivatives 180 Definitions Higher order partial The chain rule Taylor’s formula SinhVienZone.com derivatives 180 183 185 185 https://fb.com/sinhvienzonevn x Contents Functions of m variables 186 Su3cient criterion for local extrema 188 195 Nemytskii operators and the calculus of variations Nemytskii operators The continuity of Nemytskii operators The differentiability of Nemytskii operators The differentiability of parameter-dependent integrals Variational problems The Euler–Lagrange equation Classical mechanics Inverse maps 195 195 197 200 202 204 207 Zo ne 212 214 217 218 Implicit functions 221 nh Vi en Di2erentiable maps on product spaces The implicit function theorem Regular values Ordinary di2erential equations Separation of variables Lipschitz continuity and uniqueness The PicardLindelăof theorem 212 The derivative of the inverse of linear maps The inverse function theorem Di2eomorphisms The solvability of nonlinear systems of equations C om 221 223 226 226 229 233 235 Manifolds 242 Si Submanifolds of Rn Graphs The regular value theorem The immersion theorem Embeddings Local charts and parametrizations Change of charts 10 Tangents and normals SinhVienZone.com 242 243 243 244 247 252 255 260 The tangential in R The tangential space Characterization of the tangential space Di2erentiable maps The di2erential and the gradient Normals Constrained extrema Applications of Lagrange multipliers n 260 261 265 266 269 271 272 273 https://fb.com/sinhvienzonevn Contents xi Chapter VIII Line integrals Curves and their lengths 281 Curves in Rn Zo nh Vi en 281 282 284 286 292 293 294 295 298 300 300 302 303 308 310 312 314 316 317 321 Line integrals 326 Si The definition Elementary properties The fundamental theorem of line integrals Simply connected sets The homotopy invariance of line integrals Pfa2 forms 308 Vector fields and Pfa2 forms The canonical basis Exact forms and gradient fields The Poincar´e lemma Dual operators Transformation rules Modules 292 Unit tangent vectors Parametrization by arc length Oriented bases The Frenet n-frame Curvature of plane curves Identifying lines and circles Instantaneous circles along curves The vector product The curvature and torsion of space curves C om The total variation Rectifiable paths Di2erentiable curves Rectifiable curves ne 326 328 330 332 333 Holomorphic functions 339 Complex line integrals Holomorphism The Cauchy integral theorem The orientation of circles The Cauchy integral formula Analytic functions Liouville’s theorem The Fresnel integral The maximum principle SinhVienZone.com 339 342 343 344 345 346 348 349 350 https://fb.com/sinhvienzonevn 208 VII Multivariable di4erential calculus ditions  t1 L(t, q, q) ˙ dt = ⇒ Min for q C [t0 , t1 ], Rm , q(t0 ) = q0 , q(t1 ) = q1 t0 It then follows from Theorem 6.12 that q satisfies the Euler–Lagrange equation C om d (Lq˙ ) = Lq dt Zo ne if the appropriate regularity conditions are also satisfied In physics, engineering, and other fields that use variational methods (for example, economics), it is usual to presume the regularity conditions are satisfied and to postulate the validity of the Euler–Lagrange equation (and to see the existence of an extremal path as physically obvious) The Euler–Lagrange equation is then used to extract the form of the extremal path and to subsequently understand the behavior of the system en 6.14 Examples (a) We consider the motion of an unconstrained point particle of (positive) mass m moving in three dimensions acted on by a time-independent potential field U (t, q) = U (q) We assume the kinetic energy depends only on q˙ and is given by 2 T (q) ˙ = m |q| ˙ Vi The EulerLagrange di2erential equation takes the form mă q = U (q) (6.23) nh Because qă is the acceleration of the system, (6.23) is the same as Newton’s equation of motion for the motion of a particle acted on by a conservative7 force −∇U Si Proof We identify (R3 )4 = L(R3 , R) with R3 using the Riesz representation theorem We then get from L(t, q, q) ˙ = T (q) ˙ − U (q) the relations ∂2 L(t, q, q) ˙ = −∇U (q) and ∂3 L(t, q, q) ˙ = ∇T (q) ˙ = mq˙ , which prove the claim (b) In a generalization of (a), we consider the motion of N unconstrained, massive point particles in a potential field We write x = (x1 , , xN ) for the coordinates, where xj R3 specifies the position of the j-th particle Further suppose X is open in R3N and U C (X, R) Letting mj be the mass of the j-th particle, the kinetic energy of the system is T (x) ˙ := N  mj j=1 7A 2 |x˙ j | force field is called conservative if it has a potential (see Remark VIII.4.10(c)) SinhVienZone.com https://fb.com/sinhvienzonevn VII.6 Nemytskii operators and the calculus of variations 209 Then we get a system of N EulerLagrange equations: ăj = ∇xj U (x) −mj x for j4 N Here, ∇xj denotes the gradient of U in the variables xj R3 C om Exercises ne Suppose I = [α, ], where −6 < < < , k C(I × I, R), 2 C 0,1 (I × E, F ) Also let  2 Φ(u)(t) := k(t, s)2 s, u(s) ds for t I 3 and u C(I, E) Prove that C C(I, E), C(I, F ) and  2 3 ∂Φ(u)h (t) = k(t, s)∂2 s, u(s) h(s) ds for t I and u, h C(I, E) I I (Hint: Consider J5 C(I × J, E) Show J en J Zo Suppose I and J := [α, ] are compact perfect intervals and f that     f (s, t) ds dt = f (s, t) dt ds E,   y 5 I y f (s, t) dt ds nh Vi and apply Proposition 6.7 and Corollary VI.4.14.) 3 For f C [α, ], E , show  s  t  s f (τ ) dτ dt = (s − t)f (t) dt for s [α, ] Suppose I and J are compact perfect intervals and ρ C(I × J, R) Verify that   log ((x − s)2 + (y − t)2 ρ(s, t) dt ds for (x, y) R2 \(I × J) u(x, y) := Si I J is harmonic  Define h: R and, for −6