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Applied Functional Analysis Lecture Notes Fall, 2010

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Applied Functional Analysis Lecture Notes Fall, 2010 Applied Functional Analysis Lecture Notes Fall, 2010 Dr. H. T. Banks Center for Research in Scientific Computation Department of Mathematics N. C. State University August 18, 2010 1Preface 1.1 Features of These Notes In these lectures, we shall present functional analysis for partial differential equations (PDE’s) or distributed parameter systems (DPS) as the basis of modern PDE techniques.This is in contrast to classical PDE techniques such as separation of variables, Fourier transforms, Sturm-Louiville prob- lems, etc.It is also somewhat different from the emphasis in usual functional analysis courses where one learns functional analysis as a subdiscipline in its own right.Here we treat functional analysis as a tool to be used in understanding and treating distributed parameter systems.We shall also motivate our discussions with numerous application examples from biology, electromagnetics and materials/mechanics. ∙ Sometimes we give proofs, sometimes not!!–Our choice–if proof sheds light or is not readily found in literature, we might give some or all of arguments behind a result- ∙ examples are quite diverse and details given varying widely -for ex- ample, heat,, transport, beam examples are widely found and well documented in literature-don’t always give much in formulation—on other hand, static two-player min-max example not widely found, cel- lular HIV leading to delay systems, population models with aggregate data, relaxed or chattering controls vs mixing distributions vs Preisach hysteresis and relation to Prohorov metric, some of applications of FA to IP and control not so widely found– 1.2 Functional Analysis for Infinite Dimensional Systems As we shall see, functional analysis techniques can often provide powerful tools for for treatment of PDEs and FDEs in areas of: ∙ Modeling ∙ Qualitative analysis ∙ Inverse problems ∙ Feedback Control ∙ Engineering analysis 1 ∙ Approximation and Computation (such as finite element and spectral methods) 2 2 Introduction to Functional Analysis 2.1 Goals of this Course In these lectures, we shall present functional analysis for partial differential equations (PDE’s) or distributed parameter systems (DPS) as the basis of modern PDE techniques.This is in contrast to classical PDE techniques such as separation of variables, Fourier transforms, Sturm-Louiville prob- lems, etc.It is also somewhat different from the emphasis in usual functional analysis courses where one learns functional analysis as a subdiscipline in its own right.Here we treat functional analysis as a tool to be used in understanding and treating distributed parameter systems.We shall also motivate our discussions with numerous application examples from biology, electromagnetics and materials/mechanics. 2.2 Uses of Functional Analysis for PDEs As we shall see, functional analysis techniques can often provide powerful tools for insight into a number of areas including: ∙ Modeling ∙ Qualitative analysis ∙ Inverse problems ∙ Control ∙ Engineering analysis ∙ Computation (such as finite element and spectral methods) 2.3 Example 1: Heat Equation ∂ ∂ ∂ (, )= (() (, )) + (, )0<<, >0(1) ∂ ∂ ∂ where (, ) denotes the temperature in the rod at time and position and (, ) is the input from a source, e.g. heat lamp, laser, etc. B.C. =0: (, 0) = 0 Dirichlet B.C. This indicates temperature is held constant. 3 ∂ = : () ∂ (, ) = 0 Neumann B.C. This indicates an insulated end and results in heat flux being zero. ∂ Here (, )=() ∂ (, ) is called the heat flux. I.C. (0,)=Φ()0<< Here Φ denotes the initial temperature distribution in the rod. 2.4 Some Preliminary Operator Theory Let , be normed linear spaces.Then ℒ(, )={ : → ∣bounded, linear} is also a normed linear space, and ∣ ∣ℒ(, ) =sup∣∣ . ∣∣ =1 Recall that linear spaces are closed under addition and scalar multiplication. A normed linear space is complete if for any Cauchy sequence {} there exists ∈ such that lim→∞ = . Note: ℒ(, )iscompleteif is complete. If is a Hilbert or Banach space (complex), then is a bounded linear operator from 1 → 2,i.e., ∈ℒ(1,2) ⇐⇒ is continuous. Differentiation in Normed Linear Spaces [HP] Suppose : → is a (nonlinear) transformation (mapping). (0+)−(0) Definition 1 If lim→0+ exists, we say has a directional derivative at 0 in direction .This is denoted (0; ), and is called the Gateaux differential at 0 in direction . If the limit exists for any direction ,wesay is Gateaux differentiable at 0 and → (0; ) is Gateaux derivative.Note that → (0; )isnot necessarily linear, however it is homogeneous of degree one (i.e., (0; )= (0; ) for scalars ).Moreover, it need not be continuous in . The following definition of (∣∣) will be useful in defining the Fr´echet derivative: 4 ∣()∣ Definition 2 ()is(∣∣)if ∣∣ → 0as → 0. Definition 3 If there exists (0; ⋅) ∈ℒ(, ) such that ∣(0 + ) − (0) − (0; )∣ = (∣∣ ), then (0; ⋅)istheFr´echet derivative of (0+)−(0) at 0.Equivalently, lim →0 = (0; ) for every ∈ with ′ → (0; ) ∈ℒ(, ).We write (0; )= (0). Results 1.If has a Fr´echet derivative, it is unique. 2.If has a Fr´echet derivative at 0, then it also has a Gateaux derivative at 0 and they are the same. 3.If : ⊂ → has a Fr´echet derivative at 0,then is continuous at 0. Examples 1. = ℝ, = ℝ,and : → such that each component of has partials at in the form ∂ ( ).Then 0 ∂ 0 ( ) ∂ ′( )= ( ) 0 ∂ 0 and hence ( ) ∂ ( ; )= ( ) . 0 ∂ 0 2.Now we consider the case where = ℝ and = ℝ1.Then (0; )=▽(0) ⋅ . 3.We also look at the case where = = ℝ1.Here, ′ (0; )= (0). 1 ′ Normally = 1 because that is the only direction in ℝ .Then (0) ′ is called the derivative, but in actuality → (0) is the derivative. ′ 1 1 1 1 Note that (0) ∈ℒ(ℝ , ℝ ), but the elements of ℒ(ℝ , ℝ )arejust numbers. 5 Homework Exercises ∙ Ex.1 :Consider푓 : ℝ2 → ℝ1. { 2 1 2 , if 푥 ∕=0 2+4 푓(푥1,푥2)= 1 2 0, if 푥 =0 Show 푓 is Gateaux differentiable at 푥 =(푥1,푥2) = 0, but not contin- uous at 푥 = 0, (and hence cannot be Fr´echet differentiable). 2.5 Transforming the Initial Boundary Value Problem If we can transform the initial boundary value problem (IBVP) for equation (1) into something of the form { 푥˙(푡)=퐴푥(푡)+퐹 (푡) (2) 푥(0) = 푥0 then conceptually it might be an easier problem with which to work.That is, formally it looks like an ordinary differential equation problem for which 푒 is a solution operator. To rigorously make this transformation and develop a corresponding con- ceptual framework, we need to undertake several tasks: 1.Find a space 푋 of functions and an operator 퐴 : (퐴) ⊂ 푋 → 푋 such that the IBVP can be written in the form of equation (2).We may find 푋 = 퐿2(0,푙)or퐶(0,푙). We want a solution 푥(푡)=푦(푡, ⋅) to (2), but in what sense - mild, weak, strong, classical? This will answer questions about regularity (i.e., smoothness) of solutions. 2.We also want solution operators or “semigroups” 푇 (푡) which play the role of 푒.But what does “ 푒”meaninthiscase?If퐴 ∈ 푅 × 푅 is a constant matrix, then ∑∞ (퐴푡) 퐴2푡2 푒 ≡ = 퐼 + 퐴푡 + + ... 푛! 2! =0 where the series has nice convergent properties.In this case, by the “variation of constants” or “variation of parameters” representation, we can then write ∫ (−) 푥(푡)=푒 푥0 + 푒 퐹 (푠)푑푠. 0 6 We want a similar variation of constants representation in 푋 with operators 푇 (푡) ∈ℒ(푋)=ℒ(푋, 푋) such that 푇 (푡) ∼ 푒,sothat ∫ 푥(푡)=푇 (푡)푥0 + 푇 (푡 − 푠)퐹 (푠)푑푠. 0 holds and represents the PDE solutions in some appropriate sense and can be used for qualitative (stability, asymptotic behavior, control) and quantitative (approximation and numerics) analyses. 7 3 Semigroups and Infinitesimal Generators 3.1 Basic Principles of Semigroups [HP, Pa, Sh, T] Definition 4 A semigroup is a one parameter set of operators {푇 (푡): 푇 (푡) ∈ℒ(푋)},where푋 is a Banach or Hilbert space, such that 푇 (푡)satisfies 1. 푇 (푡 + 푠)=푇 (푡)푇 (푠) (semigroup or Markov or translation property) 2. 푇 (0) = 퐼.(identity property) Classification of semigroups by continuity ∙ 푇 (푡)isuniformly continuous if lim ∣푇 (푡) − 퐼∣ =0. →0+ This is not of interest to us, because 푇 (푡) is uniformly continuous if and only if 푇 (푡)=푒 where 퐴 is a bounded linear operator. ∙ 푇 (푡)isstrongly continuous, denoted 퐶0,ifforeach푥 ∈ 푋, 푡 → 푇 (푡)푥 is continuous on [0,훿) for some positive 훿. Note 1: All continuity statements are in terms of continuity from the right at zero.For fixed 푡 푇 (푡 + ℎ) − 푇 (푡)=푇 (푡)[푇 (ℎ) − 푇 (0)] = 푇 (푡)[푇 (ℎ) − 퐼] and 푇 (푡) − 푇 (푡 − 휖)=푇 (푡 − 휖)[푇 (휖) − 퐼] so that continuity from the right at zero is equivalent to continuity at any 푡 for operators that are uniformly bounded on compact intervals. Note 2: 푇 (푡) uniformly continuous implies 푇 (푡) strongly continuous (∣푇 (푡)푥 − 푥∣≤∣푇 (푡) − 퐼∣∣푥∣), but not conversely. 3.2 Return to Example 1 : Heat Equation To begin the process of writing the system of Example 1 in the form of equation (2), take 푋 = 퐿2(0,푙) and define (퐴)={휑 ∈ 퐻2(0,푙)∣휑(0) = 0,휑′(푙)=0} 8 to be the domain of 퐴.(Assume 퐷 is smooth for now, e.g., 퐷 is at least 퐻1.) Then we can define 퐴 : (퐴) ⊂ 푋 → 푋 by 퐴[휑](휉)=(퐷(휉)휑′(휉))′. (3) Notation: In general, 푊 ,(푎, 푏)={휑 ∈ 퐿∣휑′,휑′′,...,휑() ∈ 퐿}. ,2 If 푝 =2,wewrite푊 (푎, 푏)=퐻 (푎, 푏).퐴퐶(푎, 푏)={휑 ∈ 퐿2(푎, 푏)∣휑 is absolutely continuous (퐴퐶)on[푎, 푏]}. Homework Exercises ∙ Ex.2 : Show that 퐴 : (퐴) ⊂ 푋 → 푋 of the heat example (Example 1) is not bounded 푋 → 푋.
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