DOCSLIB.ORG

Introduction to Sobolev Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Introduction to Sobolev Spaces Cambridge University Press 978-1-108-41570-5 — Finite Elements Sasikumaar Ganesan , Lutz Tobiska Excerpt More Information chapter 1 ........................................................................................................ INTRODUCTION TO SOBOLEV SPACES ........................................................................................................ In thischapterwe recall some basics on functional analysisandprovideabrief introduction to Sobolev spaces. For amore detailed and comprehensive study, we refer to Adams (1975). 1.1. Banach and hilbert spaces ................................................................................................................ Definition 1.1. Let X beareallinear space. Amapping : X R iscalled anormonX if · → x 0 x 0 for all x X, • = ⇔ = ∈ λx λ x for all x X, λ R, • =| | ∈ ∈ x y x y for all x,y X. • + ≤ + ∈ The pair (X, ) iscalled anormed space. · Setting y x inthe inequality and using theother two properties weget x 0 for all =− ≥ x X. ∈ Definition 1.2. Asequence (xn)n N iscalled Cauchy sequence, iffor all ε>0thereexists an ∈ index n0(ε)such that for all m,n > n0(ε) it holds xm xn <ε. − Definition 1.3. Asequence (xn)n N converges to x X iffor all ε>0there isanindex n0(ε) ∈ ∈ such that for all n > n0(ε) it holds xn x <ε. − Definition 1.4. Anormed space (X, ) iscalled complete if every Cauchy sequence in X · converges in X. Definition 1.5. Acomplete normed space iscalledBanach space. © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-41570-5 — Finite Elements Sasikumaar Ganesan , Lutz Tobiska Excerpt More Information 2 introduction to sobolev spaces Example 1.1. Let Rd, d isthe dimension, beanopen andboundeddomain. We denote by ⊂ Lp(), 1 p < ,thesetof all measurable functions f : R forwhich ≤ ∞ → f (x) p dx < . | | ∞ Similarly,thesetof all measurable functions f : R satisfying → esssup f (x) : x < {| | ∈ } ∞ will be denotedbyL (). Then, Lp(), p [1, ] isareallinear space.Setting for 1 p < ∞ ∈ ∞ ≤ ∞ 1/p p f p : f (x) dx and f : esssup f (x) : x , L () = | | L∞( ) = {| | ∈ } respectively, Lp() becomesanormed space for any p [1, ] ifwe identify functions which ∈ ∞ areequal up toasetof measurezero. This identification is necessary because weonly have for a set M of measurezero ⊂ f (x) p dx 0 f (x) 0 for all x M. | | = ⇒ = ∈ \ Strictly speaking, theelements in Lp(), p [1, ]areclasses of equivalent functions (equal up ∈ ∞ to a set of measurezero). Then, thefirst condition in Definition 1.1 becomes true. Thesecond condition follows directly from the definition of p and thethird condition is just the ·L () inequality stated in Lemma 1.1. Lemma 1.1 (Minkowski’s inequality). For f ,g Lp(),p [1, ],wehave ∈ ∈ ∞ f g p f p g p . + L () ≤ L () + L () Theorem 1.1. Lp(),p [1, ], is a Banach space. ∈ ∞ Lemma 1.2 (Hölders’s inequality). For f Lp() and g Lq() with p,q [1, ], 1/p ∈ ∈ ∈ ∞ + 1/q 1,wehavefg L1() and = ∈ fg 1 f p g q . L () ≤ L () L ( ) Definition 1.6. Twonorms 1 and 2 onanormed space X arecalled equivalent if there · · exist positive constants C1, C2 such that C1 x 1 x 2 C2 x 1 for all x X. ≤ ≤ ∈ Note that topologicalproperties do not change when switching to an equivalent norm, for example,asequence isconvergent in(X, 1) iff it isconvergent in(X, 2). · · Definition 1.7. Let (X, X ) beanormed space. Amapping g : X R iscalledlinear if · → g(αx βy) αg(x) βg(y) for all α,β R, x,y X. + = + ∈ ∈ A linear mapping g : X R iscontinuous if there is a constant C such that → g(x) C x X for all x X. | |≤ ∈ © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-41570-5 — Finite Elements Sasikumaar Ganesan , Lutz Tobiska Excerpt More Information 1.1. banach and hilbert spaces 3 Definition 1.8. We define the sum of twocontinuous linearfunctionals, g1 and g2,and the multiplication of acontinuous linearfunctional g with a real number α by (g1 g2)(x): g1(x) g2(x)and (αg)(x): αg(x), α R, x X. + = + = ∈ ∈ Then, it can beshownthat thesetof all continuous linearfunctionalscreate a reallinear space, the dual space X . In the following, for theelements g X we use the notation g,x : g(x). ∗ ∈ ∗ = Lemma 1.3. The set X of continuous linear functionals x g,x on X is a Banach space with ∗ → respect to the norm g,x g X∗ : sup | |. = 0 x X x X = ∈ Proof. Using the definition of the dual norm X , wesee · ∗ g X 0 g,x 0 for all x X g 0. ∗ = ⇔ = ∈ ⇔ = Further, we have λg,x λ g,x g,x λg X∗ sup | | sup | | λ sup | | λ g X∗ = 0 x X x X = 0 x X x X =| | 0 x X x X =| | = ∈ = ∈ = ∈ andfor the sum of two functionals it holds g1 g2,x g1,x g2,x g1 g2 X∗ sup | + | sup | + | + = 0 x X x X = 0 x X x X = ∈ = ∈ g1,x g2,x sup | | sup | | g1 X∗ g2 X∗ ≤ 0 x X x X + 0 x X x X = + = ∈ = ∈ It remains to show that (X∗, X ) iscomplete. Let (gn)n N beaCauchy sequence in X∗,that · ∗ ∈ is, for all ε>0there isanindex n0(ε)such that for all m,n > n0(ε) it holds gm gn X <ε. − ∗ Then, for afixed x X,thesequence ofreal numbers(gn,x )n N isaCauchy sequence in R ∈ ∈ due to gm,x gn,x gm gn,x gm gn X x X <ε x X | − | = | − | ≤ − ∗ for all m,n > n0(ε). Any Cauchy sequence ofreal numbers isconvergent in R,that means limn gn,x g(x) for all x X. The linearity of g follows from →∞ = ∈ g(αx βy) lim gn,αx βy lim α gn,x β gn,y n n + = →∞ + = →∞ + α lim gn,x β lim gn,y αg(x) βg(y). n n = →∞ + →∞ = + Concerning thecontinuity of g we recall that Cauchy sequences are bounded,thus gn,x gn X x X C x X for all x X. | | ≤ ∗ ≤ ∈ The limit n showsthat the linear mapping g : X R iscontinuous. →∞ → © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-41570-5 — Finite Elements Sasikumaar Ganesan , Lutz Tobiska Excerpt More Information 4 introduction to sobolev spaces Definition 1.9. Let V bealinear space. Amapping ( , ):V V R iscalled an innerproduct · · × → on V if (u,v) (v,u) for all u,v V, • = ∈ (αu βv,w) α(u,w) β(v,w) for all u,v,w V, α,β R, • + = + ∈ ∈ (u,u) > 0 for u 0. • = AnormonV is inducedbysetting u : √(u,u). If (V, V ) iscomplete wecall V aHilbert = · space. Lemma 1.4 (Schwarz inequality). LetVbeaHilbertspace.Then, (u,v) u V v V for all u,v V. | |≤ ∈ Example 1.2. V L2()equippedwith the innerproduct = (f ,g): f (x)g(x)dx = isaHilbertspace. Theorem 1.2 (Riesz representation theorem). Let V be a Hilbert space. There is a unique linear mapping R : V V from the dual space V into the Hilbert space V such that for all g V and ∗ → ∗ ∈ ∗ v V ∈ (Rg,v) g,v and Rg V g V . = = ∗ Fixedpoint theorems areoften used to establish theunique solvability of an abstract oper- ator equation. Let us consider the following problem inaBanach space V with an operator P : V V: → Find u V such that u Pu. (1.1) ∈ = Solutions u V ofProblem (1.1) arecalled fixedpoints of themapping P : V V. ∈ → Definition 1.10. Themapping P : V V iscalled contractive or to beacontraction if there is → a positive constant ρ<1such that Pv1 Pv2 V ρ v1 v2 V for all v1,v2 V. − ≤ − ∈ Theorem 1.3 (Banach’sfixedpoint theorem). Let V be a Banach space and P : V Vbea → contraction. Then, we have the following statements: (1) There is a unique fixed point u V solving Problem (1.1). ∗ ∈ (2) The sequence un Pun 1,n 1, converges to the fixed point u∗ V for any initial guess = − ≥ ∈ u0 V. ∈ (3) It holds the error estimate ρn un u∗ V Pu0 u0 V for all n N. − ≤ 1 ρ − ∈ − © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-41570-5 — Finite Elements Sasikumaar Ganesan , Lutz Tobiska Excerpt More Information 1.2. weak derivatives 5 Proof. We start by estimating the distance of ui V to ui 1 V ∈ − ∈ 2 ui ui 1 V Pui 1 Pui 2 V ρ ui 1 ui 2 V ρ ui 2 ui 3 V − − = − − − ≤ − − − ≤ − − − . ≤ i 1 ui ui 1 V ρ − Pu0 u0 V for all i N. − − ≤ − ∈ From that wegetfor m > n 1 ≥ m m um un V (ui ui 1) ui ui 1 V − = − − ≤ − − i n 1 V i n 1 =+ =+ m m n i 1 n 1 ρ − ρ − Pu0 u0 V ρ − Pu0 u0 V ≤ − ≤ 1 ρ − i n 1 =+ − ρn Pu0 u0 V 0 for n . ≤ 1 ρ − → →∞ − Therefore, for any ε>0, thereexists an index bound n0(ε)such that for all m > n n0(ε) ≥ ρn um un V Pu0 u0 V <ε, − ≤ 1 ρ − − consequently,(un) is a Cauchy sequence and convergestosomeu V. We show that u isa ∗ ∈ ∗ fixedpoint of P. Indeed, u∗ Pu∗ V u∗ un V Pun 1 Pu∗ V − ≤ − + − − u∗ un V ρ un 1 u∗ V 0 for n . ≤ − + − − → →∞ Next we prove that there is at most one fixedpoint. Assume that therearetwofixedpoints u u . Then, weconclude 1∗ = 2∗ u∗ u∗ V Pu∗ Pu∗ V ρ u∗ u∗ V (1 ρ) u∗ u∗ V 0. 1 − 2 = 1 − 2 ≤ 1 − 2 ⇒ − 1 − 2 ≤ Since ρ<1 wegetu u . Finally, we recall the estimate for m > n 1 1∗ = 2∗ ≥ ρn un u∗ V un um V um u∗ V Pu0 u0 V um u∗ V − ≤ − + − ≤ 1 ρ − + − − from which the last statement follows by m .
Load more

Recommended publications
  • Sobolev Spaces, Theory and Applications
    Sobolev spaces, theory and applications Piotr Haj lasz1 Introduction These are the notes that I prepared for the participants of the Summer School in Mathematics in Jyv¨askyl¨a,August, 1998. I thank Pekka Koskela for his kind invitation. This is the second summer course that I delivere in Finland. Last August I delivered a similar course entitled Sobolev spaces and calculus of variations in Helsinki. The subject was similar, so it was not posible to avoid overlapping. However, the overlapping is little. I estimate it as 25%. While preparing the notes I used partially the notes that I prepared for the previous course. Moreover Lectures 9 and 10 are based on the text of my joint work with Pekka Koskela [33]. The notes probably will not cover all the material presented during the course and at the some time not all the material written here will be presented during the School. This is however, not so bad: if some of the results presented on lectures will go beyond the notes, then there will be some reasons to listen the course and at the same time if some of the results will be explained in more details in notes, then it might be worth to look at them. The notes were prepared in hurry and so there are many bugs and they are not complete. Some of the sections and theorems are unfinished. At the end of the notes I enclosed some references together with comments. This section was also prepared in hurry and so probably many of the authors who contributed to the subject were not mentioned.
    [Show full text]
  • Introduction to Sobolev Spaces
    Introduction to Sobolev Spaces Lecture Notes MM692 2018-2 Joa Weber UNICAMP December 23, 2018 Contents 1 Introduction1 1.1 Notation and conventions......................2 2 Lp-spaces5 2.1 Borel and Lebesgue measure space on Rn .............5 2.2 Definition...............................8 2.3 Basic properties............................ 11 3 Convolution 13 3.1 Convolution of functions....................... 13 3.2 Convolution of equivalence classes................. 15 3.3 Local Mollification.......................... 16 3.3.1 Locally integrable functions................. 16 3.3.2 Continuous functions..................... 17 3.4 Applications.............................. 18 4 Sobolev spaces 19 4.1 Weak derivatives of locally integrable functions.......... 19 1 4.1.1 The mother of all Sobolev spaces Lloc ........... 19 4.1.2 Examples........................... 20 4.1.3 ACL characterization.................... 21 4.1.4 Weak and partial derivatives................ 22 4.1.5 Approximation characterization............... 23 4.1.6 Bounded weakly differentiable means Lipschitz...... 24 4.1.7 Leibniz or product rule................... 24 4.1.8 Chain rule and change of coordinates............ 25 4.1.9 Equivalence classes of locally integrable functions..... 27 4.2 Definition and basic properties................... 27 4.2.1 The Sobolev spaces W k;p .................. 27 4.2.2 Difference quotient characterization of W 1;p ........ 29 k;p 4.2.3 The compact support Sobolev spaces W0 ........ 30 k;p 4.2.4 The local Sobolev spaces Wloc ............... 30 4.2.5 How the spaces relate.................... 31 4.2.6 Basic properties { products and coordinate change.... 31 i ii CONTENTS 5 Approximation and extension 33 5.1 Approximation............................ 33 5.1.1 Local approximation { any domain............. 33 5.1.2 Global approximation on bounded domains.......
    [Show full text]
  • Using Functional Analysis and Sobolev Spaces to Solve Poisson’S Equation
    USING FUNCTIONAL ANALYSIS AND SOBOLEV SPACES TO SOLVE POISSON'S EQUATION YI WANG Abstract. We study Banach and Hilbert spaces with an eye to- wards defining weak solutions to elliptic PDE. Using Lax-Milgram we prove that weak solutions to Poisson's equation exist under certain conditions. Contents 1. Introduction 1 2. Banach spaces 2 3. Weak topology, weak star topology and reflexivity 6 4. Lower semicontinuity 11 5. Hilbert spaces 13 6. Sobolev spaces 19 References 21 1. Introduction We will discuss the following problem in this paper: let Ω be an open and connected subset in R and f be an L2 function on Ω, is there a solution to Poisson's equation (1) −∆u = f? From elementary partial differential equations class, we know if Ω = R, we can solve Poisson's equation using the fundamental solution to Laplace's equation. However, if we just take Ω to be an open and connected set, the above method is no longer useful. In addition, for arbitrary Ω and f, a C2 solution does not always exist. Therefore, instead of finding a strong solution, i.e., a C2 function which satisfies (1), we integrate (1) against a test function φ (a test function is a Date: September 28, 2016. 1 2 YI WANG smooth function compactly supported in Ω), integrate by parts, and arrive at the equation Z Z 1 (2) rurφ = fφ, 8φ 2 Cc (Ω): Ω Ω So intuitively we want to find a function which satisfies (2) for all test functions and this is the place where Hilbert spaces come into play.
    [Show full text]
  • Function Spaces Mikko Salo
    Function spaces Lecture notes, Fall 2008 Mikko Salo Department of Mathematics and Statistics University of Helsinki Contents Chapter 1. Introduction 1 Chapter 2. Interpolation theory 5 2.1. Classical results 5 2.2. Abstract interpolation 13 2.3. Real interpolation 16 2.4. Interpolation of Lp spaces 20 Chapter 3. Fractional Sobolev spaces, Besov and Triebel spaces 27 3.1. Fourier analysis 28 3.2. Fractional Sobolev spaces 33 3.3. Littlewood-Paley theory 39 3.4. Besov and Triebel spaces 44 3.5. H¨olderand Zygmund spaces 54 3.6. Embedding theorems 60 Bibliography 63 v CHAPTER 1 Introduction In mathematical analysis one deals with functions which are dif- ferentiable (such as continuously differentiable) or integrable (such as square integrable or Lp). It is often natural to combine the smoothness and integrability requirements, which leads one to introduce various spaces of functions. This course will give a brief introduction to certain function spaces which are commonly encountered in analysis. This will include H¨older, Lipschitz, Zygmund, Sobolev, Besov, and Triebel-Lizorkin type spaces. We will try to highlight typical uses of these spaces, and will also give an account of interpolation theory which is an important tool in their study. The first part of the course covered integer order Sobolev spaces in domains in Rn, following Evans [4, Chapter 5]. These lecture notes contain the second part of the course. Here the emphasis is on Sobolev type spaces where the smoothness index may be any real number. This second part of the course is more or less self-contained, in that we will use the first part mainly as motivation.
    [Show full text]
  • Fact Sheet Functional Analysis
    Fact Sheet Functional Analysis Literature: Hackbusch, W.: Theorie und Numerik elliptischer Differentialgleichungen. Teubner, 1986. Knabner, P., Angermann, L.: Numerik partieller Differentialgleichungen. Springer, 2000. Triebel, H.: H¨ohere Analysis. Harri Deutsch, 1980. Dobrowolski, M.: Angewandte Funktionalanalysis, Springer, 2010. 1. Banach- and Hilbert spaces Let V be a real vector space. Normed space: A norm is a mapping k · k : V ! [0; 1), such that: kuk = 0 , u = 0; (definiteness) kαuk = jαj · kuk; α 2 R; u 2 V; (positive scalability) ku + vk ≤ kuk + kvk; u; v 2 V: (triangle inequality) The pairing (V; k · k) is called a normed space. Seminorm: In contrast to a norm there may be elements u 6= 0 such that kuk = 0. It still holds kuk = 0 if u = 0. Comparison of two norms: Two norms k · k1, k · k2 are called equivalent if there is a constant C such that: −1 C kuk1 ≤ kuk2 ≤ Ckuk1; u 2 V: If only one of these inequalities can be fulfilled, e.g. kuk2 ≤ Ckuk1; u 2 V; the norm k · k1 is called stronger than the norm k · k2. k · k2 is called weaker than k · k1. Topology: In every normed space a canonical topology can be defined. A subset U ⊂ V is called open if for every u 2 U there exists a " > 0 such that B"(u) = fv 2 V : ku − vk < "g ⊂ U: Convergence: A sequence vn converges to v w.r.t. the norm k · k if lim kvn − vk = 0: n!1 1 A sequence vn ⊂ V is called Cauchy sequence, if supfkvn − vmk : n; m ≥ kg ! 0 for k ! 1.
    [Show full text]
  • Optimal Sobolev Embeddings and Function Spaces
    Optimal Sobolev embeddings and Function Spaces Nadia Clavero Director: Javier Soria Tutor: Joan Cerd`a M´asteren Matem´aticaAvanzada y Profesional Especialidad Acad´emicaAvanzada Curso 2010/2011 U UNIVERSITAT DE BARCELONA B Contents 1 Introduction5 2 Preliminaries 11 2.1 The distribution function............................. 11 2.2 Decreasing rearrangements............................ 12 2.3 Rearrangement invariant Banach function spaces................ 16 2.4 Orlicz spaces................................... 18 2.5 Lorentz spaces................................... 22 2.6 Lorentz Zygmund spaces............................. 25 2.7 Interpolation spaces................................ 26 2.8 Weighted Hardy operators............................ 28 3 Sobolev spaces 35 3.1 Introduction.................................... 35 3.2 Definitions and basic properties......................... 35 3.3 Riesz potencials.................................. 38 3.4 Sobolev embedding theorem........................... 40 4 Orlicz spaces and Lorentz spaces 49 4.1 Introduction.................................... 49 4.2 Sobolev embeddings into Orlicz spaces..................... 50 4.3 Sobolev embeddings into Lorentz spaces.................... 52 4.4 Sobolev embeddings into Lorentz Zygmund spaces............... 56 5 Optimal Sobolev embeddings on rearrangement invariant spaces 59 5.1 Introduction.................................... 59 5.2 Reduction Theorem................................ 59 5.3 Optimal range and optimal domain of r.i. norms................ 70 5.4 Examples....................................
    [Show full text]
  • Sobolev-Type Spaces (Mainly Based on the Lp Norm) on Metric Spaces, and Newto- Nian Spaces in Particular, Have Been Under Intensive Study Since the Mid-"##$S
    - ) . / 0 &/ . ! " # $" ! %&' '( ! ) *+, !"## $ "% ' ( &)*+) ,##(-./*01*2/ ,#3(+140+)01*)+0.*/0- %"52-)/ 6 7 %80$%6!"6#62-)/ i Abstract !is thesis consists of four papers and focuses on function spaces related to !rst- order analysis in abstract metric measure spaces. !e classical (i.e., Sobolev) theory in Euclidean spaces makes use of summability of distributional gradients, whose de!nition depends on the linear structure of Rn. In metric spaces, we can replace the distributional gradients by (weak) upper gradients that control the functions’ behav- ior along (almost) all recti!able curves, which gives rise to the so-called Newtonian spaces. !e summability condition, considered in the thesis, is expressed using a general Banach function lattice quasi-norm and so an extensive framework is built. Sobolev-type spaces (mainly based on the Lp norm) on metric spaces, and Newto- nian spaces in particular, have been under intensive study since the mid-"##$s. In Paper I, the elementary theory of Newtonian spaces based on quasi-Banach function lattices is built up. Standard tools such as moduli of curve families and the Sobolev capacity are developed and applied to study the basic properties of Newto- nian functions. Summability of a (weak) upper gradient of a function is shown to guarantee the function’s absolute continuity on almost all curves. Moreover, New- tonian spaces are proven complete in this general setting. Paper II investigates the set of all weak upper gradients of a Newtonian function. In particular, existence of minimal weak upper gradients is established. Validity of Lebesgue’s di%erentiation theorem for the underlying metric measure space ensures that a family of representation formulae for minimal weak upper gradients can be found.
    [Show full text]
  • Topological Invariants for Discontinuous Mappings Differential Topology in Sobolev Spaces
    Topological invariants for discontinuous mappings Differential topology in Sobolev spaces Paweł Goldstein [email protected] Kolokwium MIM, 7 listopada 2019 TOPOLOGICAL INVARIANTS FOR DISCONTINUOUS MAPPINGS 1/21 Two reasons: • natural: folding, breaking, changes of state; • technical: necessary because of the available mathematical tools. Introduction Why do we need • non-differentiable • discontinuous mappings in real-life applications? TOPOLOGICAL INVARIANTS FOR DISCONTINUOUS MAPPINGS 2/21 Introduction Why do we need • non-differentiable • discontinuous mappings in real-life applications? Two reasons: • natural: folding, breaking, changes of state; • technical: necessary because of the available mathematical tools. TOPOLOGICAL INVARIANTS FOR DISCONTINUOUS MAPPINGS 2/21 Minima of accumulative, i.e, integral quantities ) we measure distance to minimizer with integral expressions. Introduction Variational principles • Fermat’s principle (Hero of Alexandria, Pierre de Fermat): Light travels through media along paths of shortest time. • Extremal action principle (Euler, Maupertuis): Bodies travel along paths locally minimizing the reduced action (integral of the momentum). Other examples: Dirichlet’s principle for harmonic maps, Einstein-Hilbert action in general relativity etc. TOPOLOGICAL INVARIANTS FOR DISCONTINUOUS MAPPINGS 3/21 Introduction Variational principles • Fermat’s principle (Hero of Alexandria, Pierre de Fermat): Light travels through media along paths of shortest time. • Extremal action principle (Euler, Maupertuis):
    [Show full text]
  • Characterizations of Screened Sobolev Spaces Noah Stevenson and Ian Tice DEPARTMENTOF MATHEMATICAL SCIENCES,CARNEGIE MELLON UNIVERSITY
    Characterizations of Screened Sobolev Spaces Noah Stevenson and Ian Tice DEPARTMENT OF MATHEMATICAL SCIENCES,CARNEGIE MELLON UNIVERSITY An approach which is promising in the resolution of this issue is to swap the inhomogeneous Sobolev spaces for their homogeneous counterparts, Does the topology on the screened Sobolev spaces Background W_ k;p (Ω). The upshot is that the latter families allow for a larger variety of Research admit a Fourier space characterization? behaviors at infinity. For this approach to have any chance of being fruitful in the Leoni and Tice gave the following partial frequency & Motivation study of boundary value problems, it is essential to identify and understand the Questions n characterization of these spaces: for f 2 S R it holds Partial differential equations (PDEs) are trace spaces associated to the homogeneous Sobolev spaces. Recall that the trace Z 1 2s 2 ^ 2 2 central to the modeling of natural space associated to a Sobolev space on a domain is a space of functions holding Can the screened Sobolev spaces be [f]W~ s;2 minfjξj ; jξj g f (ξ) dξ ; (2) (σ) n understood through interpolation theory? R phenomena. As a result, the study of these the possible boundary values and, for higher order Sobolev spaces, powers of the where 0 < s < 1 and constant screening function σ = 1. equations and their solutions has held a normal derivative. Interpolation theory is a set of tools which This expression suggests that the high-mode part of a prominent place in mathematics for For functions of finite energy in planar strips these trace spaces were first s;2 take certain pairs of topological vector member of W~ behaves like a member of the centuries.
    [Show full text]
  • Sobolev Spaces on an Arbitrary Metric Space
    Potential Analysis 5: 403-41 5, 1996. @ 1996 Kluwer Academic Publishers. Printed in the Netherlands. Sobolev Spaces on an Arbitrary Metric Space PIOTR HAJLASZ* lnstvtut Matematyki, Uniwersyter Warszawski ul. Banucha 2, 02-097 Warszawa, Poland ?-mail: [email protected] (Received: 7 February 1994; accepted: 6 October 1994) Abstract. We define Sobolev space JV',~for 1 < p < (x, on an arbitrary metric space with finite diameter and equipped with finite, positive Bore1 measure. In the Euclidean case it coincides with standard Sobolev space. Several classical imbedding theorems are special cases of general results which hold in the metric case. We apply our results to weighted Sobolev space with Muckenhoupt weight. Mathematics Subject Classifications (1991). Primary: 46E35; Secondary: 28A80 Key words: Generalized Sobolev spaces, imbedding theorems, Muckenhoupt weights, metric space. 1. Introduction It is well known that [li1?"consists of Lipschitz functions and hence we can naturally define LV'~" on an arbitrary metric space. In the case 1 < p 6 m we find a Lipschitz type characterization of W'J'(Theorem 1) which can be used as a definition of T/I~'>Pin the case of metric space as a domain. Since the space 1V13" has an integral nature, we have to equip the metric space with a measure. The imbedding theorems depend on the dimension; so if we want to get imbed- ding theorems in this general, metric context, we have to introduce a condition describing the dimension. This condition is very simple. It suffices to assume that p(B(x,r)) 2 CIrS (see the definition of s-regularity in Section 5).
    [Show full text]
  • Chapter 2 Sobolev Spaces 2.1 Preliminaries
    Chapter 2 Sobolev spaces In this chapter, we give a brief overview on basic results of the theory of Sobolev spaces and their associated trace and dual spaces. 2.1 Preliminaries Let ­ be a bounded domain in Euclidean space lRd. We denote by ­ its closure and refer to ¡ = @­ := ­ n ­ as its boundary. Moreover, we d denote by ­e := lR n ­ the associated exterior domain. We consider functions u : ­ ! lR and denote by j®j ® @ u D u : = ®1 ®2 ®d @x1 @x2 :::@xd T d its partial derivatives of order j®j. Here, ® = (®1; :::; ®d) 2 lN0 is a Pd multiindex of modulus j®j = i=1 ®i. m We de¯ne by C (­); m 2 lN0; the linear space of continuous functions on ­ whose partial derivatives D®u; j®j · m; exist and are continuous. Cm(­) is a Banach space with respect to the norm ® kukCm(­) := max sup jD u(x)j : 0·j®j·m x2­ m;® We refer to C (­); m 2 lN0; 0 < ® < 1; as the linear space of func- tions in Cm(­) whose m-th order partial derivatives are HÄoldercon- d tinuous, i.e., for all ¯ 2 lN0 with j¯j = m there exist constants ¡¯ > 0 such that for all x; y 2 ­ ¯ ¯ ® jD u(x) ¡ D u(y)j · ¡¯ jx ¡ yj : We note that Cm;®(­) is a Banach space with respect to the norm jD¯u(x) ¡ D¯u(y)j m;® m kukC (­) := kukC (­) + max sup ® : j¯j=m x;y2­ jx ¡ yj m m;® Moreover, C0 (­) and C0 (­) are the subspaces of functions with compact support in ­).
    [Show full text]
  • On the Dual of Bv
    ON THE DUAL OF BV MONICA TORRES Abstract. An important problem in geometric measure theory is the characterization of BV ∗, the dual of the space of functions of bounded variation. In this paper we survey recent results that show that the solvability of the equation div F = T is closely connected to the problem of characterizing BV ∗. In particular, the (signed) measures in BV ∗ can be characterized in terms of the solvability of the equation div F = T . 1. Introduction It is an open problem in geometric measure theory to give a full characterization of BV ∗, the dual of the space of functions of bounded variation (see Ambrosio-Fusco-Pallara [3, Remark 3.12]). Meyers and Ziemer characterized in [21] the positive measures in Rn that belong to the dual of BV (Rn), where BV (Rn) is the space of all functions in L1(Rn) whose distributional gradient is a vector-measure in Rn with finite total variation. They showed that the positive measure µ belongs to BV (Rn)∗ if and only if µ satisfies the condition µ(B(x; r)) ≤ Crn−1 for every open ball B(x; r) ⊂ Rn and C = C(n). Besides the classical paper by Meyers and Ziemer, we refer the interested reader to the paper by De Pauw [11], where the author analyzes SBV ∗, the dual of the space of special functions of bounded variation. In Phuc-Torres [22] we showed that there is a connection between the problem of characterizing BV ∗ and the solvability of the equation div F = T . Indeed, we showed that the (signed) measure µ belongs to BV (Rn)∗ if and only if there exists a bounded vector field F 2 L1(Rn; Rn) such that div F = µ.
    [Show full text]