Philippe. Functional Analysis with Applications. Linear and Nonlinear. G. Ciarlet. City University of Hong Kong. siajtl

Transcription

1 Philippe G Ciarlet City University of Hong Kong Linear and Nonlinear Functional Analysis with Applications with 401 Problems and 52 Figures siajtl Society for Industrial and Applied Mathematics Philadelphia

2 CONTENTS Preface xiii 1 Real Analysis and Theory of Functions: A Quick Review 1 Introduction 1 11 Sets 2 12 Mappings 3 13 The axiom of choice and Zorn's lemma 5 14 Construction of the sets E and C 8 15 Cardinal numbers; finite and infinite sets 9 16 Topological spaces Continuity in topological spaces Compactness in topological spaces Connectedness and simple-connectedness in topological spaces Metric spaces Continuity and uniform continuity in metric spaces Complete metric spaces Compactness in metric spaces The Lebesgue measure in R"; measurable functions The Lebesgue integral in Mn; the basic theorems Change of variable in Lebesgue integrals in Rn Volumes, areas, and lengths in R" The spaces Cm{n) and Cm(U); domains in R" 36 2 Normed Vector Spaces 43 Introduction Vector spaces; Hamel bases; dimension of a vector space Normed vector spaces; first properties and examples; quotient spaces The space C(K;Y) with K compact; uniform convergence and local uniform convergence The spaces p, 1 < p < oo The Lebesgue spaces LP{Sl), 1 < p < oo Regularization and approximation in the spaces LP(Q), 1 <p < oo Compactness and finite-dimensional normed vector spaces; F Riesz theorem Application of compactness in finite-dimensional normed vector spaces: The fundamental theorem of algebra 79 vii

3 viii Contents 29 Continuous linear operators in normed vector spaces; the spaces C(X;Y), L{X), and X' Compact linear operators in normed vector spaces Continuous multilinear mappings in normed vector spaces; the space Ck(Xi,X2,,Xk;Y) Korovkin's theorem Application of Korovkin's theorem to polynomial approximation; Bohman's, Bernstein's, and Weierstrafi' theorems Application of Korovkin's theorem to trigonometric polynomial approximation; Fejer's theorem The Stone-Weierstrafi theorem Convex sets Convex functions Banach Spaces 123 Introduction Banach spaces; first properties First examples of Banach spaces; the spaces C(K; Y) with K compact and Y complete, and C(X; Y) with Y complete Integral of a continuous function of a real variable with values in a Banach space Further examples of Banach spaces: the spaces P and 1^(0), 1 < p < oo Dual of a normed vector space; first examples; F Riesz representation theorem in LP{Sl), 1 < p < oo Series in Banach spaces Banach fixed point theorem Application of Banach fixed point theorem: Existence of solutions to nonlinear ordinary differential equations; Cauchy-Lipschitz theorem; the pendulum equation Application of Banach fixed point theorem: Existence of solutions to nonlinear two-point boundary value problems Ascoli-Arzela's theorem Application of Ascoli-Arzela's theorem: Existence of solutions to nonlinear ordinary differential equations; Cauchy-Peano theorem; Euler's method Inner-Product Spaces and Hilbert Spaces 173 Introduction Inner-product spaces and Hilbert spaces; first properties; Cauchy-Schwarz-Bunyakovskii inequality; parallelogram law First examples of inner-product spaces and Hilbert spaces; the spaces 2 and L2(Q) The projection theorem Application of the projection theorem: Least-squares solution of a linear system Orthogonality; direct sum theorem 195

4 Contents ix 46 F Riesz representation theorem in a Hilbert space First applications of the F Riesz representation theorem: Hahn-Banach theorem in a Hilbert space; adjoint operators; reproducing kernels Maximal orthonormal families in an inner-product space Hilbert bases and Fourier series in a Hilbert space Eigenvalues and eigenvectors of self-adjoint operators in inner-product spaces The spectral theorem for compact self-adjoint operators The "Great Theorems" of Linear Functional Analysis 231 Introduction Baire's theorem; a first application: Noncompleteness of the space of all polynomials Application of Baire's theorem: Existence of nowhere differentiable continuous functions Banach-Steinhaus theorem, alias the uniform boundedness principle; application to numerical quadrature formulas Application of the Banach-Steinhaus theorem: Divergence of Lagrange interpolation Application of the Banach-Steinhaus theorem: Divergence of Fourier series Banach open mapping theorem; a first application: Well-posedness of two- 272 point boundary value problems Banach closed graph theorem; a first application: Hellinger-Toeplitz theorem The Hahn-Banach theorem in a vector space The Hahn-Banach theorem in a normed vector space; first consequences Geometric forms of the Hahn-Banach theorem; separation of convex sets 511 Dual operators; Banach closed range theorem Weak convergence and weak * convergence Banach-Saks-Mazur theorem Reflexive spaces; the Banach-Eberlein-Smulian theorem Linear Partial Differential Equations 305 Introduction Quadratic minimization problems; variational equations and variational inequalities The Lax-Milgram lemma Weak partial derivatives in Lloc( l); a brief incursion into distribution theory Hypoellipticity of A The Sobolev spaces Wm'p{Cl) and Hm{n): First properties The Sobolev spaces Wm>p{0) and Hm{Q) with Q a domain; imbedding theorems, traces, Green's formulas Examples of second-order linear elliptic boundary value problems; the membrane problem Examples of fourth-order linear boundary value problems; the biharmonic and plate problems 355

5 Contents 69 Examples of nonlinear boundary value problems associated with variational inequalities; obstacle problems Eigenvalue problems for second-order elliptic operators The spaces W_m-«(n) and #-m(fi); JL Lions lemma The Babuska-Brezzi inf-sup theorem; application to constrained quadratic minimization problems Application of the Babuska-Brezzi inf-sup theorem: Primal, mixed, and dual formulations of variational problems Application of the Babuska-Brezzi inf-sup theorem and of JL Lions lemma: The Stokes equations A second application of JL Lions lemma: Korn's inequality Application of Korn's inequality: The equations of three-dimensional linearized elasticity The classical PoincarS lemma and its weak version as an application of JL Lions lemma and of the hypoellipticity of A Application of Poincar^'s lemma: The classical and weak Saint-Venant lemmas; the Cesaro-Volterra path integral formula Another application of JL Lions lemma: The Donati lemmas Pfaff systems 444 Differential Calculus in Normed Vector Spaces 451 Introduction The Frechet derivative; the chain rule; the Piola identity; application to extrema of real-valued functions The mean value theorem in a normed vector space; first applications Application of the mean value theorem: Differentiability of the limit of a sequence of differentiable functions Application of the mean value theorem: Differentiability of a function defined by an integral Application of the mean value theorem: Sard's theorem A mean value theorem for functions of class C1 with values in a Banach space Newton's method for solving nonlinear equations; the Newton-Kantorovich theorem in a Banach space Higher order derivatives; Schwarz lemma Taylor formulas; application to extrema of real-valued functions Application: Maximum principle for second-order linear elliptic operators Application: Lagrange interpolation in R" and multipoint Taylor formulas Convex functions and differentiability; application to extrema of real-valued functions The implicit function theorem; first application: Class C of the mapping A^A' The local inversion theorem; the invariance of domain theorem for mappings of class C1 in Banach spaces; class C of the mapping A -> Ax/ Constrained extrema of real-valued functions; Lagrange multipliers Lagrangians and saddle-points; primal and dual problems 565

6 Contents xi Differential Geometry in Rn 575 Introduction Curvilinear coordinates in an open subset of Rn Metric tensor; volumes and lengths in curvilinear coordinates Covariant derivative of a vector field Tensors a brief introduction Necessary conditions satisfied by the metric tensor; the Riemann curvature tensor Existence of an immersion on an open subset of Rn with a prescribed metric tensor; the fundamental theorem of Riemannian geometry Uniqueness up to isometries of immersions with the same metric tensor; the rigidity theorem for an open subset of Rn Curvilinear coordinates on a surface in R First fundamental form of a surface; areas, lengths, and angles on a surface Isometric, equiareal, and conformal surfaces Second fundamental form of a surface; curvature on a surface Principal curvatures; Gaussian curvature Covariant derivatives of a vector field defined on a surface; the Gaufi and Weingarten formulas Necessary conditions satisfied by the first and second fundamental forms: The Gaufi and Codazzi-Mainardi equations Gaufi Theorema Egregium; application to cartography Existence of a surface with prescribed first and second fundamental forms; the fundamental theorem of surface theory Uniqueness of surfaces with the same fundamental forms; the rigidity theorem for surfaces 654 The "Great Theorems" of Nonlinear Functional Analysis 657 Introduction Nonlinear partial differential equations as the Euler-Lagrange equations associated with the minimization of a functional Convex functions and sequentially lower semicontinuous functions with values in R U {oo} Existence of minimizers for coercive and sequentially weakly lower semicontinuous functionals Application to the von Karman equations Existence of minimizers in!v1,p(f2) Application to the p-laplace operator Polyconvexity; compensated compactness; John Ball's existence theorem in nonlinear elasticity Ekeland's variational principle; existence of minimizers for functionals that satisfy the Palais-Smale condition Brouwer's fixed point theorem a first proof Application of Brouwer's theorem to the von Karman equations, by means of the Galerkin method 726

7 xii Contents 911 Application of Brouwer's theorem to the Navier-Stokes equations, by means of the Galerkin method Schauder's fixed point theorem; Schafer's fixed point theorem; Leray-Schauder fixed point theorem Monotone operators The Minty-Browder theorem for monotone operators; application to the p-laplace operator The Brouwer topological degree in K": Definition and properties Brouwer's fixed point a second proof and the hairy ball theorem 764 theorem 917 Borsuk's and Borsuk-Ulam theorems; Brouwer's invariance of domain theorem 767 Bibliographical Notes 777 Bibliography 781 Main Notations 807 Index 815