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1985 | Buch

Introduction to the Subject Kapitel lesen Erstes Kapitel lesen

Nonlinear Functional Analysis and its Applications

III: Variational Methods and Optimization

verfasst von: Eberhard Zeidler

Verlag: Springer New York

Enthalten in: Professional Book Archive

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As long as a branch of knowledge offers an abundance of problems, it is full of vitality. David Hilbert Over the last 15 years I have given lectures on a variety of problems in nonlinear functional analysis and its applications. In doing this, I have recommended to my students a number of excellent monographs devoted to specialized topics, but there was no complete survey-type exposition of nonlinear functional analysis making available a quick survey to the wide range of readers including mathematicians, natural scientists, and engineers who have only an elementary knowledge of linear functional analysis. I have tried to close this gap with my five-part lecture notes, the first three parts of which have been published in the Teubner-Texte series by Teubner-Verlag, Leipzig, 1976, 1977, and 1978. The present English edition was translated from a completely rewritten manuscript which is significantly longer than the original version in the Teubner-Texte series. The material is organized in the following way: Part I: Fixed Point Theorems. Part II: Monotone Operators. Part III: Variational Methods and Optimization. Parts IV jV: Applications to Mathematical Physics. The exposition is guided by the following considerations: (a) What are the supporting basic ideas and what intrinsic interrelations exist between them? (/3) In what relation do the basic ideas stand to the known propositions of classical analysis and linear functional analysis? ( y) What typical applications are there? Vll Preface viii Special emphasis is placed on motivation.

Inhaltsverzeichnis

Frontmatter

Introduction to the Subject

Introduction to the Subject
Abstract
Extremal problems play an extraordinarily large role in the application of mathematics to practical problems, for example:
(α)
in mathematical physics (mechanics and celestial mechanics, geometrical optics, elasticity theory, hydrodynamics, rheology, relativity theory, etc.);
 
(β)
in geometry (geodesics, minimal surfaces, etc.);
 
(γ)
in mathematical economics (transport problems, optimal warehouse maintenance);
 
(δ)
in regulation technology (optimal control of general regulation systems, e.g., industrial installations, spaceships, etc.);
 
(ε)
in chemistry, geophysics, technology, etc. (optimal determination of unknown data from measurements);
 
(ζ)
in numerical mathematics (optimal structuring of approximation processes, etc.);
 
(η)
in the theory of probability (optimal control of stochastic processes, optimal estimation of unknown parameters, optimal construction of airplanes, water-power networks, etc.).
 
Eberhard Zeidler

Introductory Typical Examples

Chapter 37. Introductory Typical Examples
Abstract
In the following we wish to present many concrete examples, foregoing extensive technical details, whose solutions have contributed essentially to the development of a general theory of extremal problems. A glance at the organization of this chapter in the Contents shows the variety of different problems one encounters. In this connection, an especially central position is assumed by Section 37.4, where we discuss a number of fundamental ideas from the classical calculus of variations. The ideas of the calculus of variations have influenced the modern theory of extremal problems in an essential way, and knowledge of these classical ideas is indispensable for a thorough understanding of the modern development.
Eberhard Zeidler

Two Fundamental Existence and Uniqueness Principles

Frontmatter
Chapter 38. Compactness and Extremal Principles
Abstract
In this chapter we give a far-reaching generalization of the following classical theorem of Weierstrass using compactness arguments: A continuous function F: [a, b] → ℝ, − oo < a < b < ∞, has a maximum and a minimum (see Fig. 38.1). Here, lower semicontinuous functionals and weak sequentially lower semicontinuous functionals play a crucial role. In this connection, we exploit, e.g., the fact that the continuity of F: [a, b] → ℝ is not needed for the existence of a minimum of F, but only the lower semicontinuity.
Eberhard Zeidler
Chapter 39. Convexity and Extremal Principles
Abstract
In the preceding chapter we showed how existence propositions for extremal problems are obtained with the aid of compactness arguments. A second basic strategy for obtaining existence propositions consists in considering convexity instead of compactness. Figure 39.1 shows the logical connections. We place the Hahn-Banach theorem at the pinnacle; in the final analysis this theorem goes back to the central fixed point theorem of Bourbaki and Kneser, in Chapter 11, via Zorn’s lemma. The separation theorems for convex sets and the Krein extension theorem for positive functionals follow from the Hahn-Banach theorem. These three theorems are standard results of functional analysis. We summarize them in Section 39.1 without proofs. The proofs can be found, e.g., in Edwards (1965, M). In fact these three theorems, which are framed in Fig. 39.1, are mutually equivalent if they are appropriately formulated. They represent different conceptions of a general fundamental principle of geometric functional analysis, which finds its most suggestive geometrical form in the separation theorems for convex sets. These equivalences are discussed in Holmes (1975, M), page 95 (cf. Problem 39.13).
Eberhard Zeidler

Extremal Problems without Side Conditions

Frontmatter
Chapter 40. Free Local Extrema of Differentiable Functionals and the Calculus of Variations
Abstract
In this chapter, in an elementary way, we generalize the known classical criteria, mentioned in Section 37.1, for free local extrema of differentiate real functions to functionals. Theorem 40.A in Section 40.2 forms the foundation of the classical calculus of variations.
Eberhard Zeidler
Chapter 41. Potential Operators
Abstract
Together with the minimum problem
(1)
we consider the Euler equation
(2)
and ask the following questions:
(α)
How can one obtain the solutions for (2) from the solutions of (1)?
 
(β)
Which operator equations Au−b = 0 can be written in the form (2), i.e., when is F′ = A?
 
(γ)
In what manner are the properties of F connected with those of F′?
 
Eberhard Zeidler
Chapter 42. Free Minima for Convex Functionals, Ritz Method and the Gradient Method
Abstract
In this chapter we show the intimate connection between the convexity of the functional F and the monotonicity of the operator F′ which fully corresponds to the known connection in the case of real functions F: ℝ → ℝ. In this way we obtain an approach to the theory of monotone operators F′ by means of convex minimum problems. In contrast to general minimum problems, convex minimum problems have a number of crucial advantages:
(i)
According to the main theorem and its variants in Sections 38.3 and 38.5, there result simple existence propositions in reflexive B-spaces.
 
(ii)
By Theorem 38.C, it follows from the strict convexity of F that the minimum point is unique.
 
(iii)
Local minima are always global minima.
 
(iv)
The Euler equation F′(u) = 0, where u ∈ int D(F), is not only a necessary condition but also a sufficient condition for a free local minimum of F at u.
 
(v)
One has productive approximation methods at one’s disposal in the Ritz and gradient methods.
 
Eberhard Zeidler

Extremal Problems with Smooth Side Conditions

Frontmatter
Chapter 43. Lagrange Multipliers and Eigenvalue Problems
Abstract
In this chapter we shall show what nondegeneracy condition is necessary to justify the Lagrange multiplier rule in the narrower sense for smooth side conditions. Moreover, we will interpret this condition geometrically and explain the connection with manifolds in B-spaces. In this connection, a generalization of the implicit function theorem is the focal point (Theorem 43.C). The central concepts are:
(α)
Tangent vector, tangent space, and submersion.
 
(β)
Regular point of a set.
 
(γ)
Manifold.
 
(δ)
Tangential mapping.
 
(ε)
Critical point of a functional.
 
Eberhard Zeidler
Chapter 44. Ljusternik-Schnirelman Theory and the Existence of Several Eigenvectors
Abstract
In Chapter 43 we proved the existence of an eigenvector. Now we concern ourselves with the eigenvalue problem
(1)
and we will prove the existence of several eigenvectors for (1) within the generalized context of the Courant maximum-minimum principle. In this connection, in an essential way, we use the fact that A and B are odd potential operators, i.e., A = F′, B = G′, and A(− μ) = − A(μ), B(−μ) = − B(μ) for all μ ∈ X. We have already explained the basic idea of the Ljusternik-Schnirelman theory in Section 37.26, and we recommend that the reader first study Section 37.26 again.
Eberhard Zeidler
Chapter 45. Bifurcation for Potential Operators
Abstract
In this chapter we shall show that especially favorable bifurcation relations are present in the case of potential operators. In the proof of the main theorem, we shall essentially make use of Lagrange multipliers and the Ljusternik-Schnirelman theory. We introduced the basic concepts of bifurcation theory in Section 8.1.
Eberhard Zeidler

Extremal Problems with General Side Conditions

Frontmatter
Chapter 46. Differentiable Functionals on Convex Sets
Abstract
In this chapter, by generalizing the results of Section 40.2, we show that in the case of a convex set M each solution u of
https://static-content.springer.com/image/chp%3A10.1007%2F978-1-4612-5020-3_11/978-1-4612-5020-3_11_Equa_HTML.gif
If u is an interior point of M, then, by Problem 39.4, (2) passes to the Euler equation
https://static-content.springer.com/image/chp%3A10.1007%2F978-1-4612-5020-3_11/978-1-4612-5020-3_11_Equb_HTML.gif
Eberhard Zeidler
Chapter 47. Convex Functionals on Convex Sets and Convex Analysis
Abstract
Over the last 20 years, parallel to the theory of monotone operators, a calculus for the investigation of convex functionals designated by convex analysis has emerged, which allows one to solve a number of problems in a simple way. To this calculus belong:
(α)
The subgradient ∂F (a generalization of the classical concept of derivative).
 
(β)
The conjugate functional F* (duality theory).
 
Eberhard Zeidler
Chapter 48. General Lagrange Multipliers (Dubovickii-Miljutin Theory)
Abstract
In Chapter 43 (eigenvalue problems) and in Section 47.10 (Kuhn-Tucker theory), we became acquainted with the Lagrange multiplier method for handling extremal problems. In this chapter we prove a very general formulation of this method (Theorem 48.A in Section 48.3). In this connection, the direction cone and the positive functionals that exist on it play the crucial role.
Eberhard Zeidler

Saddle Points and Duality

Frontmatter
Chapter 49. General Duality Principle by Means of Lagrange Functions and Their Saddle Points
Abstract
In this chapter we set Lagrange functions and a related general duality principle at the pinnacle of duality theory. We treat important examples of Lagrange functions in:
(α)
Section 49.3 (linear optimization).
 
(β)
Section 50.1 (Kuhn-Tucker theory).
 
(γ)
Section 51.6 (Trefftz duality for linear elliptic partial differential equations).
 
(δ)
Section 51.7 (quasilinear elliptic partial differential equations).
 
Eberhard Zeidler
Chapter 50. Duality and the Generalized Kuhn-Tucker Theory
Abstract
In this chapter we consider convex minimum problems with a finite or infinite number of side conditions and their generalizations. It turns out that the results of the classical Kuhn-Tucker theory can be carried over completely to this situation.
Eberhard Zeidler
Chapter 51. Duality, Conjugate Functionals, Monotone Operators and Elliptic Differential Equations
Abstract
In Chapter 49 we showed how one arrives at general duality propositions knowing a Lagrange function L. In this chapter, given a functional F, we define a so-called conjugate functional F*, and in Section 51.4 we explain how one can construct a Lagrange function for a given convex minimum problem with respect to F by means of F*. In this connection, the generalized Young inequality
(1a)
(1b)
and the relation
(2)
play a crucial role. One can use
(3)
or the more general relation
(3a)
to calculate F*. Together with the propositions furnished in Chapter 47 concerning subgradients, the calculus of conjugate functionals, whose principal parts are comprised in (la)–(3a), form the “crossing frog” of convex analysis. [For readers who are not familiar with railways, a “crossing frog” is a device on railroad tracks for keeping cars on the proper rails at intersections or switches.]
Eberhard Zeidler
Chapter 52. General Duality Principle by Means of Perturbed Problems and Conjugate Functionals
Abstract
In Section 37.10 we observed the following general principle for linear optimization problems in ℝ N :
The consistency of (P) and (P*) implies that (P) and (P*) are solvable.
Eberhard Zeidler
Chapter 53. Conjugate Functionals and Orlicz Spaces
Abstract
In this chapter, we consider the Orlicz spaces L H and L H* as generalizations of the Lebesgue spaces L p and L q respectively, where p, q > 1, p −1 + q −1 = 1 and explain the connection with conjugate functionals. Orlicz spaces were introduced by Orlicz in 1932.
Eberhard Zeidler

Variational Inequalities

Frontmatter
Chapter 54. Elliptic Variational Inequalities
Abstract
We consider the variational inequality
(5)
for u ∈ M and, parallel to this, the multivalued operator equation
(6)
under the following assumptions:
Eberhard Zeidler
Chapter 55. Evolution Variational Inequalities of First Order in H-Spaces
Abstract
In this chapter we generalize the results of Chapter 31 to problems of the form
in an H-space. In this connection we use the results that we established in Chapter 23 (generalized derivatives and evolution triples) and Chapter 32 (maximal monotone operators). As an important methodological tool, we make use of the Yosida approximation. In Chapter 66 we shall consider applications to plasticity theory.
Eberhard Zeidler
Chapter 56. Evolution Variational Inequalities of Second Order in H-Spaces
Abstract
The second-order equation
https://static-content.springer.com/image/chp%3A10.1007%2F978-1-4612-5020-3_21/MediaObjects/978-1-4612-5020-3_21_Equa_HTML.gif
can be transformed by means of the substitution u′ = v into a first-order equation
https://static-content.springer.com/image/chp%3A10.1007%2F978-1-4612-5020-3_21/MediaObjects/978-1-4612-5020-3_21_Equb_HTML.gif
Eberhard Zeidler
Chapter 57. Accretive Operators and Multivalued First-Order Evolution Equations in B-Spaces
Abstract
In this chapter, in a manner parallel to Chapter 55, we study the initial value problem
(1)
where u(t) lies in a real B-space X and A is multivalued.
Eberhard Zeidler
Backmatter
Metadaten
Titel
Nonlinear Functional Analysis and its Applications
verfasst von
Eberhard Zeidler
Copyright-Jahr
1985
Verlag
Springer New York
Electronic ISBN
978-1-4612-5020-3
Print ISBN
978-1-4612-9529-7
DOI
https://doi.org/10.1007/978-1-4612-5020-3