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1997 | Buch

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Parametrized Measures and Variational Principles

verfasst von: Pablo Pedregal

Verlag: Birkhäuser Basel

Buchreihe : Progress in Nonlinear Differential Equations and Their Applications

Enthalten in: Professional Book Archive

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Weak convergence is a basic tool of modern nonlinear analysis because it enjoys the same compactness properties that finite dimensional spaces do: basically, bounded sequences are weak relatively compact sets. Nonetheless, weak conver gence does not behave as one would desire with respect to nonlinear functionals and operations. This difficulty is what makes nonlinear analysis much harder than would normally be expected. Parametrized measures is a device to under stand weak convergence and its behavior with respect to nonlinear functionals. Under suitable hypotheses, it yields a way of representing through integrals weak limits of compositions with nonlinear functions. It is particularly helpful in comprehending oscillatory phenomena and in keeping track of how oscilla tions change when a nonlinear functional is applied. Weak convergence also plays a fundamental role in the modern treatment of the calculus of variations, again because uniform bounds in norm for se quences allow to have weak convergent subsequences. In order to achieve the existence of minimizers for a particular functional, the property of weak lower semicontinuity should be established first. This is the crucial and most delicate step in the so-called direct method of the calculus of variations. A fairly large amount of work has been devoted to determine under what assumptions we can have this lower semicontinuity with respect to weak topologies for nonlin ear functionals in the form of integrals. The conclusion of all this work is that some type of convexity, understood in a broader sense, is usually involved.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

The historical problem of the calculus of variations and of the theory of optimization is that of finding minimizers of functionals in the form of integrals defined over infinite dimensional spaces. Historically, these problems were tackled and in many instances solved through the associated Euler-Lagrange equation, which is the analogue of the critical point condition for functions defined over finite dimensional spaces. This condition usually leads to an equation or system of ordinary differential equations or partial differential equations. The search for minimizers (or in general extremals) was reduced in this way to finding certain solutions of differential equations associated to the corresponding functional. Whenever these solutions could be found explicitly or shown to exist, one would establish, under suitable assumptions, the existence of solutions to the variational principle. This way of proceeding is especially fruitful in one dimension, when the Euler-Lagrange equation is an ordinary differential equation or system. In higher dimensions nonlinear partial differential equations need to be solved and in general it is not an easy task to show existence of solutions. Consequently, attention was focussed on finding extremals directly from the functional itself: the direct method of the calculus of variations was the outcome. This method has been so successful that today it is one of the usual ways of showing existence of solutions to many nonlinear elliptic partial differential equations.

Pablo Pedregal

Chapter 2. Some Variational Problems

Abstract

This chapter gathers a collection of problems for which the analysis does not involve any differential constraint, or if it does it is in a somewhat elementary way. It is a good way of practicing with the general ideas we will pursue for more complicated situations in subsequent chapters. For this reason we do not pretend to give the sharpest hypotheses under which theorems can be proved or improved, but rather focus on understanding the main techniques in each example. Some formal proofs are left to the reader as exercises. The same principle explains why in some of the problems we do not pursue the proof of all the steps and lemmas used when they are not relevant to our discussion. Three of the examples refer to variational principles or optimization. The last one does not. This has been included with the sole purpose of providing an illustration of how some analysis in terms of parametrized measures can also be helpful and provide some insight even though the problem is not directly related to variational principles but it is placed in a completely different context: large time behavior of complicated turbulent systems.

Pablo Pedregal

Chapter 3. The Calculus of Variations under Convexity Assumptions

Abstract

The central focus of the calculus of variations is the functional

$$ I\left( u \right) = \int {_\Omega\phi \left( {x,u(x),\nabla u(x)} \right)dx} $$

where the integrand ϕ explicitly depends upon the gradient variable ∇ u.Ωis assumed to be an open, regular, bounded domain of R ^N.The admissible functions u:Ω → R ^m belong to some reflexive Sobolev space and they may satisfy some other restriction like having the boundary values prescribed.The integrand ϕ:Ω × R ^m × M ^{m × N} → R * is assumed to be a Carathéodory function. By this we simply mean that φ is measurable on thexvariable and continuous with respect to u and ∇ u. We may eventually let ϕ take on the value +∞ as indicated by R*=R∪ { +∞ }.

Pablo Pedregal

Chapter 4. Nonconvexity and Relaxation

Abstract

We have seen in the previous chapter that the weak lower semicontinuity property is crucial in order to employ the direct method of the calculus of variations to find minimizers of variational principles. This property is inherited by functionals whose integrands enjoy the appropriate convexity. Nonetheless, for an ever increasing number of interesting problems these convexity properties fail. In some cases, specific techniques may provide solutions to problems. In some others, this lack of convexity is a precursor of nonsolvability, at least in a classical sense. In the latter, highly oscillatory phenomena are usually involved. Parametrized measures were originally introduced by Young to account for oscillations in nonconvex optimal control problems where one could not reasonably expect classical solutions.

Pablo Pedregal

Chapter 5. Phase Transitions and Microstructure

Abstract

We have tried to emphasize in the previous chapter the importance of the study of variational principles for which some lack of convexity leads one to consider the behavior of minimizing sequences. From the mathematical point of view, there are two ways to proceed whenever there are no minimizers as a consequence of this lack of convexity. One is to “convexify” the energy density itself or the nonconvex constraints involved in order to obtain a new functional which can be analyzed through the techniques dicussed in Chapter 3. The task is to relate the information concerning this convexified functional with the original one. Relaxation theorems refer to this issue. Another possibility is to enlarge sufficiently the class of competing objects in some kind of generalized variational setting as to include minimizers. These generalized objects are parametrized measures. They were introduced by Young in this same context to understand ill-posed variational problems.

Pablo Pedregal

Chapter 6. Parametrized Measures

Abstract

This chapter is devoted to general issues related to parametrized measures. For this reason it is of a technical nature. We start by establishing a rather general existence theorem that can be applied to most of the situations one encounters in practice. This existence theorem provides a representation of weak limits, when they exist, of any composition with the sequence under consideration in terms of the parametrized measure associated to such a sequence (or possibly to some subsequence). It is important to stress that this result does not guarantee in any way that the weak limit exists. This is something to be obtained independently. If we have weak limits then they can be represented by an appropriate integral against the parametrized measure. If the weak limit does not exist (because of concentration effects) then parametrized measures yield a different type of information that we have yet to fully understand.

Pablo Pedregal

Chapter 7. Analysis of Parametrized Measures

Abstract

In this chapter we shall analyze more closely parametrized measures and introduce the basic tools to deal with these families of probability measures. Some of these will be used several times later. Our main goal here is to characterize parametrized measures: we are interested in knowing when a given family of probability measures can actually be generated as the parametrized measure by some sequence of functions. At this stage we do not place any further restriction on the sequences we would like to consider except for boundedness in some L^pΩ.In this regard we place ourselves in the context of Section 2 of Chapter 2. As a matter of fact, the main theorem of this chapter, Theorem 7.7, can be proved directly taking advantage of the analysis carried out there and extending it to the casepfinite by means of some technicalities involving truncation operators. This will actually be our approach to pass from p = ∞to finitepin Chapter 8 under the gradient constraint. Nonetheless we have chosen to proceed in a different way with the idea in mind of preparing some of the main techniques for the analysis of gradient parametrized measures pursued in Chapter 8.

Pablo Pedregal

Chapter 8. Analysis of Gradient Parametrized Measures

Abstract

In variational principles we are especially interested in integrands depending on gradients. For this reason we would like to study weak convergence associated to sequences of gradients. Parametrized measures associated to sequences of gradients are called gradient parametrized measures. In particular, we would like to prove a characterization for parametrized measures coming from a sequence of gradients.

Pablo Pedregal

Chapter 9. Quasiconvexity and Rank-one Convexity

Abstract

The motivation for this chapter is two-fold. On the one hand, since Jensen’s inequality has played a prominent role in our approach to weak lower semicontinuity, our analysis would be somehow incomplete without any reference to this inequality with respect to rank-one convex functions. Because quasiconvexity implies rank-one convexity, probability measures satisfying Jensen’s inequality with respect to the class of rank-one convex functions are indeed examples of gradient parametrized measures. It turns out that this family of probability measures can be understood. at least conceptually, in a nice constructive way. They are called laminates to emphasize its layering structure. As a matter of fact, laminates are almost the only way to produce explicitly examples of gradient parametrized measures. It is true that the Riemann-Lebesgue lemma allows one to consider gradient parametrized measures associated with periodic gradients. The problem is that we do not know how to decide whether they are laminates or not. The importance of laminates in the description of some equilibrium states for crystals has been stressed in Chapter 5. They are also important in the theory of composite materials and homogenization.

Pablo Pedregal

Chapter 10. Analysis of Divergence-Free Parametrized Measures

Abstract

The question we address in this chapter is the characterization of parametrized measures coming from sequences of vector-valued functions u _j Ω ⊂ R _N → R _m uniformly bounded in L ^∞(Ω) for which we have additional information in the form

$$ \left\{ {Au_j } \right\} relatively compact in H^{ - 1} \left( \Omega\right) $$

(10-1)

, for A a differential operator of type

$${{\left( {Au} \right)}_{i}} = \sum\limits_{{l,k}} {ailk\frac{{\partial {{u}^{l}}}}{{\partial {{x}_{k}}}},\;\;\;i = 1, \ldots ,s,}$$

with constant coefficientsa _ilkIn the previous chapters, we have concentrated on the fundamental case when A = curl, m replaced by m × N,

$$curl\;u = \frac{{\partial u_{l}^{i}}}{{\partial {{x}_{k}}}} - \frac{{\partial u_{k}^{i}}}{{\partial {{x}_{1}}}},\;\;i = 1, \ldots ,m.\;\;l,k = 1, \ldots ,N,$$

and we asked forAu =0 rather than (10-1). In general, let

$$ \vartheta= \left\{ {(\lambda ,\xi ) \in R^m\times R^N :\sum\limits_{l,k} {a_{ilk} \lambda _l } \xi _k= 0} \right\} $$

and let the characteristic cone be defined by

$$ \Lambda= \left\{ {\lambda\in R^m :\:there\:is\:a\:\xi\in R^N- \{ 0\} ,(\lambda ,\xi ) \in \vartheta } \right\} $$

Pablo Pedregal

Backmatter

Titel: Parametrized Measures and Variational Principles
verfasst von: Pablo Pedregal
Verlag: Birkhäuser Basel
Electronic ISBN: 978-3-0348-8886-8
Print ISBN: 978-3-0348-9815-7
DOI: https://doi.org/10.1007/978-3-0348-8886-8

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Chapter 2. Some Variational Problems

Chapter 3. The Calculus of Variations under Convexity Assumptions

Chapter 4. Nonconvexity and Relaxation

Chapter 5. Phase Transitions and Microstructure

Chapter 6. Parametrized Measures

Chapter 7. Analysis of Parametrized Measures

Chapter 8. Analysis of Gradient Parametrized Measures

Chapter 9. Quasiconvexity and Rank-one Convexity

Chapter 10. Analysis of Divergence-Free Parametrized Measures

Backmatter