Lebedev L.P., Vorovich I.I., Cloud M.J. Functional Analysis in Mechanics

2nd edition. — Springer Science+Business Media, LLC, 2013. 316 p. — ISSN: 1439-7382, 978-1-4614-5867-8, 978-1-4614-5868-5 (eBook).
This book offers a brief, practically complete, and relatively simple introduction to functional analysis. It also illustrates the application of functional analytic methods to the science of continuum mechanics. Abstract but powerful mathematical notions are tightly interwoven with physical ideas in the treatment of nontrivial boundary value problems for mechanical objects. This second edition includes more extended coverage of the classical and abstract portions of functional analysis. Taken together, the first three chapters now constitute a regular text on applied functional analysis. This potential use of the book is supported by a significantly extended set of exercises with hints and solutions. A new appendix, providing a convenient listing of essential inequalities and imbedding results, has been added. The book should appeal to graduate students and researchers in physics, engineering, and applied mathematics.Metric, Banach, and Hilbert Spaces.
Preliminaries.
H¨ older’s Inequality and Minkowski’s Inequality.
Metric Spaces of Functions.
Some Relations for the Metrics in ˜ Lp(Ω) and p.
Metrics in Energy Spaces.
Sets in a Metric Space.
Convergence in a Metric Space.
Completeness.
Completion Theorem.
Lebesgue Integrals and the Space Lp(Ω).
Banach Spaces.
Norms on n-Dimensional Spaces.
Other Examples of Banach Spaces.
Hilbert Spaces.
Factor Spaces.
Separability.
Compactness, Hausdorff Criterion.
Arzel` a’s Theorem and Its Applications.
Theory of Approximation in a Normed Space.
Decomposition Theorem, Riesz Representation.
Linear Operators and Functionals.
Space of Linear Continuous Operators.
Uniform Boundedness Theorem.
Banach–Steinhaus Principle.
Closed Operators and the Closed Graph Theorem.
InverseOperator.
Lax–Milgram Theorem.
Open Mapping Theorem.
Dual Spaces.
Hahn–Banach Theorem.
Consequences of the Hahn–Banach Theorem.
Contraction Mapping Principle.
Topology, Weak and Weak* Topologies.
Weak Topology in a Normed Space X.
Conclusion and Further Reading.
Mechanics Problems from the Functional Analysis Viewpoint.
Introduction to Sobolev Spaces.
Operator of Imbedding.
Some Energy Spaces.
Generalized Solutions in Mechanics.
Existence of Energy Solutions to Some Mechanics Problems.
OperatorFormulationofanEigenvalueProblem.
ProblemofElastico-Plasticity;SmallDeformations.
BasesandCompleteSystems;FourierSeries.
Weak Convergence in a Hilbert Space.
Ritz and Bubnov–Galerkin Methods.
Curvilinear Coordinates, Nonhomogeneous Boundary Conditions.
Bramble–HilbertLemmaandItsApplications.
Some Spectral Problems of Mechanics.AdjointOperator.
Compact Operators.
Compact Operators in Hilbert Space.
Functions Taking Values in a Banach Space.
Spectrum of a Linear Operator.
Resolvent Set of a Closed Linear Operator.
Spectrum of a Compact Operator in Hilbert Space.
Analytic Nature of the Resolvent of a Compact Linear Operator.
Spectrum of a Holomorphic Compact Operator Function.
Self-Adjoint Compact Linear Operator in Hilbert Space.
Some Applications of Spectral Theory.
Courant’sMinimaxPrinciple.
Elements of Nonlinear Functional Analysis.
Fr´ echet and Gˆ ateaux Derivatives.
Liapunov–Schmidt Method.
Critical Points of a Functional.
Von K´ arm´ anEquationsofaPlate.
Buckling of a Thin Elastic Shell.
Nonlinear Equilibrium Problem for an Elastic Shallow Shell.
Degree Theory.
Steady-State Flow of a Viscous Liquid.
Summary of Inequalities and Imbeddings.
A.1 Inequalities.
A.2 Imbeddings.
Hints for Selected Problems.In Memoriam: Iosif I. Vorovich.