Theory of Sobolev Multipliers


 Kathlyn Hicks
 2 years ago
 Views:
Transcription
1 Vladimir G. Maz'ya Tatyana O. Shaposhnikova Theory of Sobolev Multipliers With Applications to Differential and Integral Operators ^ Springer
2 Introduction Part I Description and Properties of Multipliers 1 Trace Inequalities for Functions in Sobolev Spaces Trace Inequalities for Functions in w anc * W The Case m = l The Case m > Trace Inequalities for Functions in u; and W, p > Preliminaries The (p, m)capacity Estimate for the Integral of Capacity of a Set Bounded by a Level Surface Estimates for Constants in Trace Inequalities Other Criteria for the Trace Inequality (1.2.29) with p > The Fefferman and Phong Sufficient Condition Estimate for the LqNorm with respect to an Arbitrary Measure The case l<p<q The case q<p< n/m 30 2 Multipliers in Pairs of Sobolev Spaces Introduction Characterization of the Space M(W? > W{) Characterization of the Space M(W > W l p) for p > Another Characterization of the Space M(W > W^) for 0 < I < m, pm <n,p>l Characterization of the Space M{W > W l p) for pm > n, p> 1 47
3 VI Contents OneSided Estimates for Norms of Multipliers in the Case pm <n Examples of Multipliers The Space M(W {RD f Wj(R )) Extension from a HalfSpace The Case p > The Case p = The Space M(W m p > W k p ) The Space M(W » W l q) Certain Properties of Multipliers The Space Af (iu > w l p) Multipliers in Spaces of Functions with Bounded Variation The Spaces Mbv and MBV 66 3 Multipliers in Pairs of Potential Spaces Trace Inequality for Bessel and Riesz Potential Spaces Properties of Bessel Potential Spaces Properties of the (p, m)capacity Main Result Description of M(H > H l p) Auxiliary Assertions Imbedding of M{H * H l p) into M(H < f L p ) Estimates for Derivatives of a Multiplier Multiplicative Inequality for the Strichartz Function Auxiliary Properties of the Bessel Kernel Gi Upper Bound for the Norm of a Multiplier Lower Bound for the Norm of a Multiplier Description of the Space M(H > H l p) Equivalent Norm in M(# > H l p) Involving the Norm in L mp/(m _,) Characterization of M(# * H' p ), m > I, Involving the Norm in Li, un if The Space MiH +H l p)iormp>n OneSided Estimates for the Norm in M(# > H l p) Lower Estimate for the Norm in M(#^ > H l p) Involving Morrey Type Norms Upper Estimate for the Norm in MiH > H l p) Involving Marcinkiewicz Type Norms Upper Estimates for the Norm in M(H > H l p) Involving Norms in H l n. 98 n 3.4 Upper Estimates for the Norm in M{H p * H l p) by Norms in Besov Spaces Auxiliary Assertions Properties of the Space B% x 103
4 VII Estimates for the Norm in M(# > H p ) by the Norm in B^ Estimate for the Norm of a Multiplier in MH l p{r l ) by the qvariation Miscellaneous Properties of Multipliers in M(ff > H l p) Ill 3.6 Spectrum of Multipliers in H l l p and H pl Preliminary Information Facts from Nonlinear Potential Theory Main Theorem Proof of Theorem The Space M(/i > h l p) Positive Homogeneous Multipliers The Space M(ff n p (asi) > H l p(dbi)) Other Normalizations of the Spaces h and H Positive Homogeneous Elements of the Spaces M{h + h l p) and MiH » H l p) 130 The Space M(B » B l p) with p > Introduction Properties of Besov Spaces Survey of Known Results Properties of the Operators S) Pi i and D Pi i Pointwise Estimate for Bessel Potentials Proof of Theorem Estimate for the Product of First Differences Trace Inequality for Bp, p > Auxiliary Assertions Concerning M(B > B l p) Lower Estimates for the Norm in M(B » B l p) Proof of Necessity in Theorem Proof of Sufficiency in Theorem The Case mp>n Lower and Upper Estimates for the Norm in MiB? ^B l p) Sufficient Conditions for Inclusion into M{W » W l p) with Noninteger m and / Conditions Involving the Space B ioo Conditions Involving the Fourier Transform Conditions Involving the Space B l qp Conditions Involving the Space H l n, m Composition Operator on M(W > W p ) 174
5 VIII Contents 5 The Space M(B > B[) Trace Inequality for Functions in B[(R n ) Auxiliary Facts Main Result Properties of Functions in the Space Bf (K n ) Trace and Imbedding Properties Auxiliary Estimates for the Poisson Operator Descriptions of M(JBJ" > B[) with Integer / A Norm in M(Bf > B[) Description of M(B^ > B[) Involving D h i M(J5J n (M") > 5((R n )) as the Space of Traces Interpolation Inequality for Multipliers Description of the Space M{B^ > B[) with Noninteger / Further Results on Multipliers in Besov and Other Function Spaces Peetre's Imbedding Theorem Related Results on Multipliers in Besov and TriebelLizorkin Spaces Multipliers in BMO Maximal Algebras in Spaces of Multipliers Introduction Pointwise Interpolation Inequalities for Derivatives Inequalities Involving Derivatives of Integer Order Inequalities Involving Derivatives of Fractional Order Maximal Banach Algebra in M(W * W l p) The Case p > Maximal Banach Algebra in M{W^ > W[) Maximal Algebra in Spaces of Bessel Potentials Pointwise Inequalities Involving the Strichartz Function Banach Algebra A ' Imbeddings of Maximal Algebras Essential Norm and Compactness of Multipliers Auxiliary Assertions TwoSided Estimates for the Essential Norm. The Case m > I Estimates Involving Cutoff Functions Estimate Involving Capacity (The Case mp < n, p > 1) Estimates Involving Capacity (The Case mp = n, p > 1) Proof of Theorem Sharpening of the Lower Bound for the Essential Norm in the Case m > I, mp < n, p > 1 262
6 7.2.6 Estimates of the Essential Norm for mp > n, p > 1 and for p = OneSided Estimates for the Essential Norm The Space of Compact Multipliers TwoSided Estimates for the Essential Norm in the Case m = I Estimate for the Maximum Modulus of a Multiplier in W p by its Essential Norm Estimates for the Essential Norm Involving Cutoff Functions (The Case Ip < n, p > 1) Estimates for the Essential Norm Involving Capacity (The Case Ip < n, p > 1) TwoSided Estimates for the Essential Norm in the Cases Ip > n, p > 1, and p=l Essential Norm in MW l p 281 Traces and Extensions of Multipliers Introduction Multipliers in Pairs of Weighted Sobolev Spaces in R Characterization of M{W*'P > W < a ) Auxiliary Estimates for an Extension Operator Pointwise Estimates for Tj and VT Weighted L p Estimates for r 7 and vr Trace Theorem for the Space M{W^0 * W^a) The Case / < The Case I > Proof of Theorem for I > Traces of Multipliers on the Smooth Boundary of a Domain MW p (R n ) as the Space of Traces of Multipliers in the k Weighted Sobolev Space W p J3 (R n+m ) Preliminaries ' A Property of Extension Operator Trace and Extension Theorem for Multipliers Extension of Multipliers from R n to R" Application to the First Boundary Value Problem in a HalfSpace Traces of Functions in MW p (R n+m ) on R n Auxiliary Assertions Trace and Extension Theorem Multipliers in the Space of Bessel Potentials as Traces of Multipliers Bessel Potentials as Traces An Auxiliary Estimate for the Extension Operator T MH' p as a Space of Traces 322 IX
7 Sobolev Multipliers in a Domain, Multiplier Mappings and Manifolds Multipliers in a Special Lipschitz Domain Special Lipschitz Domains Auxiliary Assertions Description of the Space of Multipliers Extension of Multipliers to the Complement of a Special Lipschitz Domain Multipliers in a Bounded Domain Domains with Boundary in the Class C 0 ' Auxiliary Assertions Description of Spaces of Multipliers in a Bounded Domain with Boundary in the Class C o> Essential Norm and Compact Multipliers in a Bounded Lipschitz Domain The Space ML p {Q) for an Arbitrary Bounded Domain Change of Variables in Norms of Sobolev Spaces (p, ODiffeomorphisms More on (p, 2)Diffeomorphisms A Particular (p,!)diffeomorphism (p, I)Manifolds Mappings T '! of One Sobolev Space into Another Implicit Function Theorems The Space M(Wp(Q) > W l p{q)) Auxiliary Results Description of the Space M(W" (tf) > W p (fi)) 369 Part II Applications of Multipliers to Differential and Integral Operators 10 Differential Operators in Pairs of Sobolev Spaces The Norm of a Differential Operator: W » W^~ k Coefficients of Operators Mapping W p into W p ~ k as Multipliers A Counterexample Operators with Coefficients Independent of Some Variables Differential Operators on a Domain Essential Norm of a Differential Operator Fredholm Property of the Schrodinger Operator Domination of Differential Operators in R n 387
8 11 Schrodinger Operator and M(w^ > w^1) Introduction Characterization of M{w\ > w^1) and the Schrodinger Operator om« A Compactness Criterion Characterization of M{W\ » Wf 1 ) Characterization of the Space M{wl((2) > w^1^)) SecondOrder Differential Operators Acting from w^ to w^ Relativistic Schrodinger Operator and M(Wj 1/2 » Wz 1/2 ) Auxiliary Assertions Main Result Corollaries of the Form Boundedness Criterion and Related Results Multipliers as Solutions to Elliptic Equations The Dirichlet Problem for the Linear SecondOrder Elliptic Equation in the Space of Multipliers Bounded Solutions of Linear Elliptic Equations as Multipliers Introduction The Case 0 > The Case f3 = Solutions as Multipliers from Wl w{p) {(2) into W^X(G) Solvability of Quasilinear Elliptic Equations in Spaces of Multipliers Scalar Equations in Divergence Form Systems in Divergence Form Dirichlet Problem for Quasilinear Equations in Divergence Form Dirichlet Problem for Quasilinear Equations in Nondivergence Form Coercive Estimates for Solutions of Elliptic equations in Spaces of Multipliers The Case of Operators in E n Boundary Value Problem in a HalfSpace On the LooNorm in the Coercive Estimate Smoothness of Solutions to Higher Order Elliptic Semilinear Systems Composition Operator in Classes of Multipliers Improvement of Smoothness of Solutions to Elliptic Semilinear Systems 477 XI
9 XII Contents 14 Regularity of the Boundary in L p Theory of Elliptic Boundary Value Problems Description of Results Change of Variables in Differential Operators Fredholm Property of the Elliptic Boundary Value Problem Boundaries in the Classes MJT 1/p, W p ~ 1/p, and M p  1/P (6) A Priori L p Estimate for Solutions and Other Properties of the Elliptic Boundary Value Problem Auxiliary Assertions Some Properties of the Operator T Properties of the Mappings A and x Invariance of the Space W p n W p Under a Change of Variables The Space W~ k for a Special Lipschitz Domain Auxiliary Assertions on Differential Operators in Divergence Form Solvability of the Dirichlet Problem in W l p{q) Generalized Formulation of the Dirichlet Problem A Priori Estimate for Solutions of the Generalized Dirichlet Problem Solvability of the Generalized Dirichlet Problem The Dirichlet Problem Formulated in Terms of Traces Necessity of Assumptions on the Domain A Domain Whose Boundary is in M% /2 D C 1 but does not Belong to M 3/2 2 (<5) Necessary Conditions for Solvability of the Dirichlet Problem Boundaries of the Class M p ~ 1/p (6) Local Characterization of M p " 1/v {5) Estimates for a Cutoff Function Description of M p ~ 1/p (J) Involving a Cutoff Function Estimate for s x Estimate for s Estimate for s Multipliers in the Classical Layer Potential Theory for Lipschitz Domains Introduction Solvability of Boundary Value Problems in Weighted Sobolev Spaces (p, k, a)diffeomorphisms Weak Solvability of the Dirichlet Problem Main Result 542
10 XIII 15.3 Continuity Properties of Boundary Integral Operators Proof of Theorems and Proof of Theorem Proof of Theorem Properties of Surfaces in the Class M p (5) Sharpness of Conditions Imposed on dq Necessity of the Inclusion dfl W p in Theorem Sharpness of the Condition dq B ( xp Sharpness of the Condition dfl M*'{5) in Theorem Sharpness of the Condition df2 e M e p{5) in Theorem Extension to Boundary Integral Equations of Elasticity Applications of Multipliers to the Theory of Integral Operators Convolution Operator in Weighted I<2Spaces Calculus of Singular Integral Operators with Symbols in Spaces of Multipliers Continuity in Sobolev Spaces of Singular Integral Operators with Symbols Depending oni Function Spaces Description of the Space M(H m <» » #' ") Main Result Corollaries 588 References 591 List of Symbols 605 Author and Subject Index 607
FRACTIONAL INTEGRALS AND DERIVATIVES. Theory and Applications
FRACTIONAL INTEGRALS AND DERIVATIVES Theory and Applications Stefan G. Samko Rostov State University, Russia Anatoly A. Kilbas Belorussian State University, Minsk, Belarus Oleg I. Marichev Belorussian
More informationAsymptotic Analysis of Fields in MultiStructures
Asymptotic Analysis of Fields in MultiStructures VLADIMIR KOZLOV Department of Mathematics, Linkoeping University, Sweden VLADIMIR MAZ'YA Department of Mathematics, Linkoeping University, Sweden ALEXANDER
More informationA MIXED TYPE IDENTIFICATION PROBLEM RELATED TO A PHASEFIELD MODEL WITH MEMORY
Guidetti, D. and Lorenzi, A. Osaka J. Math. 44 (27), 579 613 A MIXED TYPE IDENTIFICATION PROBLEM RELATED TO A PHASEFIELD MODEL WITH MEMORY DAVIDE GUIDETTI and ALFREDO LORENZI (Received January 23, 26,
More informationINTRODUCTION TO FOURIER ANALYSIS AND WAVELETS
#. INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS Mark A. Pinsky Northwestern University BROOKS/COLE * THOMSON LEARNING Australia Canada Mexico Singapore Spain United Kingdom United States 1 FOURIER SERIES
More informationFixed Point Theory. With 14 Illustrations. %1 Springer
Andrzej Granas James Dugundji Fixed Point Theory With 14 Illustrations %1 Springer Contents Preface vii 0. Introduction 1 1. Fixed Point Spaces 1 2. Forming New Fixed Point Spaces from Old 3 3. Topological
More informationof Functional Analysis and Applications Group
FIRST ANNUAL WORKSHOP TALKS: May 15, 2010, Room Sousa Pinto, 08:15 AM Luís Castro Anabela Silva Alberto Simões Ana Paula Nolasco Saburou Saitoh Anabela Ramos António Caetano Alexandre Almeida Sofia Lopes
More informationA PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS. In memory of RouHuai Wang
A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS XAVIER CABRÉ, MANEL SANCHÓN, AND JOEL SPRUCK In memory of RouHuai Wang 1. Introduction In this note we consider semistable
More informationElementary Differential Equations
Elementary Differential Equations EIGHTH EDITION Earl D. Rainville Late Professor of Mathematics University of Michigan Phillip E. Bedient Professor Emeritus of Mathematics Franklin and Marshall College
More informationEXISTENCE OF SOLUTIONS TO NONLINEAR, SUBCRITICAL HIGHERORDER ELLIPTIC DIRICHLET PROBLEMS WOLFGANG REICHEL AND TOBIAS WETH
EXISTENCE OF SOLUTIONS TO NONLINEAR, SUBCRITICAL HIGHERORDER ELLIPTIC DIRICHLET PROBLEMS WOLFGANG REICHEL AND TOBIAS WETH Abstract. We consider the th order elliptic boundary value problem Lu = f(x,
More informationHAROLD CAMPING i ii iii iv v vi vii viii ix x xi xii 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52
More informationUNIVERSITETET I OSLO
NIVERSITETET I OSLO Det matematisknaturvitenskapelige fakultet Examination in: Trial exam Partial differential equations and Sobolev spaces I. Day of examination: November 18. 2009. Examination hours:
More informationApplied Linear Algebra I Review page 1
Applied Linear Algebra Review 1 I. Determinants A. Definition of a determinant 1. Using sum a. Permutations i. Sign of a permutation ii. Cycle 2. Uniqueness of the determinant function in terms of properties
More informationA COMMENT ON LEASTSQUARES FINITE ELEMENT METHODS WITH MINIMUM REGULARITY ASSUMPTIONS
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volume 10, Number 4, Pages 899 903 c 2013 Institute for Scientific Computing and Information A COMMENT ON LEASTSQUARES FINITE ELEMENT METHODS WITH
More informationLECTURE NOTES: FINITE ELEMENT METHOD
LECTURE NOTES: FINITE ELEMENT METHOD AXEL MÅLQVIST. Motivation The finite element method has two main strengths... Geometry. Very complex geometries can be used. This is probably the main reason why finite
More informationContents. Gbur, Gregory J. Mathematical methods for optical physics and engineering digitalisiert durch: IDS Basel Bern
Preface page xv 1 Vector algebra 1 1.1 Preliminaries 1 1.2 Coordinate System invariance 4 1.3 Vector multiplication 9 1.4 Useful products of vectors 12 1.5 Linear vector Spaces 13 1.6 Focus: periodic media
More informationExtremal equilibria for reaction diffusion equations in bounded domains and applications.
Extremal equilibria for reaction diffusion equations in bounded domains and applications. Aníbal RodríguezBernal Alejandro VidalLópez Departamento de Matemática Aplicada Universidad Complutense de Madrid,
More informationOptimal boundary control of quasilinear elliptic partial differential equations: theory and numerical analysis
Optimal boundary control of quasilinear elliptic partial differential equations: theory and numerical analysis vorgelegt von Dipl.Math. Vili Dhamo von der Fakultät II  Mathematik und Naturwissenschaften
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationAn Introduction to the NavierStokes InitialBoundary Value Problem
An Introduction to the NavierStokes InitialBoundary Value Problem Giovanni P. Galdi Department of Mechanical Engineering University of Pittsburgh, USA Rechts auf zwei hohen Felsen befinden sich Schlösser,
More informationSchneps, Leila; Colmez, Coralie. Math on Trial : How Numbers Get Used and Abused in the Courtroom. New York, NY, USA: Basic Books, 2013. p i.
New York, NY, USA: Basic Books, 2013. p i. http://site.ebrary.com/lib/mcgill/doc?id=10665296&ppg=2 New York, NY, USA: Basic Books, 2013. p ii. http://site.ebrary.com/lib/mcgill/doc?id=10665296&ppg=3 New
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationNumerical Recipes in C
2008 AGIInformation Management Consultants May be used for personal purporses only or by libraries associated to dandelon.com network. Numerical Recipes in C The Art of Scientific Computing Second Edition
More informationIntroduction to the Finite Element Method
Introduction to the Finite Element Method 09.06.2009 Outline Motivation Partial Differential Equations (PDEs) Finite Difference Method (FDM) Finite Element Method (FEM) References Motivation Figure: cross
More information1. Introduction. O. MALI, A. MUZALEVSKIY, and D. PAULY
Russ. J. Numer. Anal. Math. Modelling, Vol. 28, No. 6, pp. 577 596 (2013) DOI 10.1515/ rnam20130032 c de Gruyter 2013 Conforming and nonconforming functional a posteriori error estimates for elliptic
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n1 x n1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of
More informationMATH 590: Meshfree Methods
MATH 590: Meshfree Methods Chapter 7: Conditionally Positive Definite Functions Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2010 fasshauer@iit.edu MATH 590 Chapter
More informationIndiana State Core Curriculum Standards updated 2009 Algebra I
Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and
More informationExample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x
Lecture 4. LaSalle s Invariance Principle We begin with a motivating eample. Eample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum Dynamics of a pendulum with friction can be written
More informationPublikationsliste. 1 Referierte Zeitschriftenartikel
Publikationsliste 1 Referierte Zeitschriftenartikel [1 ] An estimate for the maximum of solutions of parabolic equations with the Venttsel condition, Vestnik Leningrad. Univ. (Ser. Mat. Mekh. Astronom.,
More informationStatistical Rules of Thumb
Statistical Rules of Thumb Second Edition Gerald van Belle University of Washington Department of Biostatistics and Department of Environmental and Occupational Health Sciences Seattle, WA WILEY AJOHN
More informationNumerical Analysis An Introduction
Walter Gautschi Numerical Analysis An Introduction 1997 Birkhauser Boston Basel Berlin CONTENTS PREFACE xi CHAPTER 0. PROLOGUE 1 0.1. Overview 1 0.2. Numerical analysis software 3 0.3. Textbooks and monographs
More informationChapter 2 Limits Functions and Sequences sequence sequence Example
Chapter Limits In the net few chapters we shall investigate several concepts from calculus, all of which are based on the notion of a limit. In the normal sequence of mathematics courses that students
More informationDimension Theory for Ordinary Differential Equations
Vladimir A. Boichenko, Gennadij A. Leonov, Volker Reitmann Dimension Theory for Ordinary Differential Equations Teubner Contents Singular values, exterior calculus and Lozinskiinorms 15 1 Singular values
More informationMATH MathematicsNursing. MATH Remedial Mathematics IBusiness & Economics. MATH Remedial Mathematics IIBusiness and Economics
MATH 090  MathematicsNursing MATH 091  Remedial Mathematics IBusiness & Economics MATH 094  Remedial Mathematics IIBusiness and Economics MATH 095  Remedial Mathematics IScience (3 CH) MATH 096
More informationNotes on weak convergence (MAT Spring 2006)
Notes on weak convergence (MAT4380  Spring 2006) Kenneth H. Karlsen (CMA) February 2, 2006 1 Weak convergence In what follows, let denote an open, bounded, smooth subset of R N with N 2. We assume 1 p
More informationReference: Introduction to Partial Differential Equations by G. Folland, 1995, Chap. 3.
5 Potential Theory Reference: Introduction to Partial Differential Equations by G. Folland, 995, Chap. 3. 5. Problems of Interest. In what follows, we consider Ω an open, bounded subset of R n with C 2
More informationINVESTIGATIONS IN ANALYSIS, DIFFERENTIAL EQUATIONS AND APPLICATIONS
International Conference Advances and Perspectives of Basic Sciences in Caucasus and Central Asian Region Tbilisi, November 13, 2011 MATHEMATICS IN GEORGIA, PART 2: INVESTIGATIONS IN ANALYSIS, DIFFERENTIAL
More informationMarch 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More informationSECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS
SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31
More information1. Periodic Fourier series. The Fourier expansion of a 2πperiodic function f is:
CONVERGENCE OF FOURIER SERIES 1. Periodic Fourier series. The Fourier expansion of a 2πperiodic function f is: with coefficients given by: a n = 1 π f(x) a 0 2 + a n cos(nx) + b n sin(nx), n 1 f(x) cos(nx)dx
More informationA mixed FEM for the quadcurl eigenvalue problem
Noname manuscript No. (will be inserted by the editor) A mixed FEM for the quadcurl eigenvalue problem Jiguang Sun Received: date / Accepted: date Abstract The quadcurl problem arises in the study of
More informationDEGREE THEORY E.N. DANCER
DEGREE THEORY E.N. DANCER References: N. Lloyd, Degree theory, Cambridge University Press. K. Deimling, Nonlinear functional analysis, Springer Verlag. L. Nirenberg, Topics in nonlinear functional analysis,
More informationMATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.
MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar
More information(Refer Slide Time: 01:1101:27)
Digital Signal Processing Prof. S. C. Dutta Roy Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture  6 Digital systems (contd.); inverse systems, stability, FIR and IIR,
More informationParabolic Equations. Chapter 5. Contents. 5.1.2 WellPosed InitialBoundary Value Problem. 5.1.3 Time Irreversibility of the Heat Equation
7 5.1 Definitions Properties Chapter 5 Parabolic Equations Note that we require the solution u(, t bounded in R n for all t. In particular we assume that the boundedness of the smooth function u at infinity
More informationChapter 5: Application: Fourier Series
321 28 9 Chapter 5: Application: Fourier Series For lack of time, this chapter is only an outline of some applications of Functional Analysis and some proofs are not complete. 5.1 Definition. If f L 1
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationMATHEMATICAL METHODS OF STATISTICS
MATHEMATICAL METHODS OF STATISTICS By HARALD CRAMER TROFESSOK IN THE UNIVERSITY OF STOCKHOLM Princeton PRINCETON UNIVERSITY PRESS 1946 TABLE OF CONTENTS. First Part. MATHEMATICAL INTRODUCTION. CHAPTERS
More informationContents. Bibliografische Informationen digitalisiert durch
1 Introduction 1 1.1 Introduction to Maple 1 1.1.1 Getting Started with Maple 1 1.1.2 Plotting with Maple 3 1.1.3 Solving Linear and Nonlinear Equations 5 1.1.4 Matrix Operations 6 1.1.5 Differential Equations
More informationExistence de solutions à support compact pour un problème elliptique quasilinéaire
Existence de solutions à support compact pour un problème elliptique quasilinéaire Jacques Giacomoni, Habib Mâagli, Paul Sauvy Université de Pau et des Pays de l Adour, L.M.A.P. Faculté de sciences de
More informationMICROLOCAL ANALYSIS OF THE BOCHNERMARTINELLI INTEGRAL
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 00029939(XX)00000 MICROLOCAL ANALYSIS OF THE BOCHNERMARTINELLI INTEGRAL NIKOLAI TARKHANOV AND NIKOLAI VASILEVSKI
More informationMaximal Regularity in Weighted Spaces, Nonlinear Boundary Conditions, and Global Attractors
Maximal Regularity in Weighted Spaces, Nonlinear Boundary Conditions, and Global Attractors Zur Erlangung des akademischen Grades eines DOKTORS DER NATURWISSENSCHAFTEN von der Fakultät für Mathematik des
More informationMean value theorem, Taylors Theorem, Maxima and Minima.
MA 001 Preparatory Mathematics I. Complex numbers as ordered pairs. Argand s diagram. Triangle inequality. De Moivre s Theorem. Algebra: Quadratic equations and expressions. Permutations and Combinations.
More informationClass Meeting # 1: Introduction to PDEs
MATH 18.152 COURSE NOTES  CLASS MEETING # 1 18.152 Introduction to PDEs, Fall 2011 Professor: Jared Speck Class Meeting # 1: Introduction to PDEs 1. What is a PDE? We will be studying functions u = u(x
More information3.1 Solving Systems Using Tables and Graphs
Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system
More informationIdeal Class Group and Units
Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals
More informationIllPosed Problems in Probability and Stability of Random Sums. Lev B. Klebanov, Tomasz J. Kozubowski, and Svetlozar T. Rachev
IllPosed Problems in Probability and Stability of Random Sums By Lev B. Klebanov, Tomasz J. Kozubowski, and Svetlozar T. Rachev Preface This is the first of two volumes concerned with the illposed problems
More informationPIECEWISE LINEAR FINITE ELEMENT METHODS ARE NOT LOCALIZED
MATHEMATICS OF COMPUTATION Volume 73, Number 247, Pages 1195 1 S 255718(3)15849 Article electronically published on July 14, 23 PIECEWISE LINEAR FINITE ELEMENT METHODS ARE NOT LOCALIZED ALAN DEMLOW Abstract.
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationConstruction of the Real Line 2 Is Every Real Number Rational? 3 Problems Algebra of the Real Numbers 7
About the Author v Preface to the Instructor xiii WileyPLUS xviii Acknowledgments xix Preface to the Student xxi 1 The Real Numbers 1 1.1 The Real Line 2 Construction of the Real Line 2 Is Every Real Number
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationNumerical concepts and error analysis for elliptic Neumann boundary control problems with pointwise state and control constraints
Numerical concepts and error analysis for elliptic Neumann boundary control problems with pointwise state and control constraints Vom Fachbereich Mathematik der Universität DuisburgEssen (Campus Duisburg)
More informationAdvanced Algebra 2. I. Equations and Inequalities
Advanced Algebra 2 I. Equations and Inequalities A. Real Numbers and Number Operations 6.A.5, 6.B.5, 7.C.5 1) Graph numbers on a number line 2) Order real numbers 3) Identify properties of real numbers
More informationRieszFredhölm Theory
RieszFredhölm Theory T. Muthukumar tmk@iitk.ac.in Contents 1 Introduction 1 2 Integral Operators 1 3 Compact Operators 7 4 Fredhölm Alternative 14 Appendices 18 A AscoliArzelá Result 18 B Normed Spaces
More informationSolvability of Fractional Dirichlet Problems with Supercritical Gradient Terms.
Solvability of Fractional Dirichlet Problems with Supercritical Gradient Terms. Erwin Topp P. Universidad de Santiago de Chile Conference HJ2016, Rennes, France May 31th, 2016 joint work with Gonzalo Dávila
More information6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )
6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a nonempty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points
More informationA RIGOROUS AND COMPLETED STATEMENT ON HELMHOLTZ THEOREM
Progress In Electromagnetics Research, PIER 69, 287 304, 2007 A RIGOROU AND COMPLETED TATEMENT ON HELMHOLTZ THEOREM Y. F. Gui and W. B. Dou tate Key Lab of Millimeter Waves outheast University Nanjing,
More informationSchneps, Leila; Colmez, Coralie. Math on Trial : How Numbers Get Used and Abused in the Courtroom. New York, NY, USA: Basic Books, p i.
New York, NY, USA: Basic Books, 2013. p i. http://site.ebrary.com/lib/mcgill/doc?id=10665296&ppg=2 New York, NY, USA: Basic Books, 2013. p iii. http://site.ebrary.com/lib/mcgill/doc?id=10665296&ppg=4 New
More informationThe Conference Call Search Problem in Wireless Networks
The Conference Call Search Problem in Wireless Networks Leah Epstein 1, and Asaf Levin 2 1 Department of Mathematics, University of Haifa, 31905 Haifa, Israel. lea@math.haifa.ac.il 2 Department of Statistics,
More informationSome stability results of parameter identification in a jump diffusion model
Some stability results of parameter identification in a jump diffusion model D. Düvelmeyer Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany Abstract In this paper we discuss
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationHeat conduction problem of an evaporating liquid wedge
T. Bárta, V. Janeček, D. Pražák Heat conduction problem of an evaporating liquid wedge MATHKMA2014/472 October 2014 Submitted Department of Mathematical Analysis, Faculty of Mathematics and Physics,
More informationCONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian. Pasquale Candito and Giuseppina D Aguí
Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian Pasquale
More informationMoving Least Squares Approximation
Chapter 7 Moving Least Squares Approimation An alternative to radial basis function interpolation and approimation is the socalled moving least squares method. As we will see below, in this method the
More informationWeak topologies. David Lecomte. May 23, 2006
Weak topologies David Lecomte May 3, 006 1 Preliminaries from general topology In this section, we are given a set X, a collection of topological spaces (Y i ) i I and a collection of maps (f i ) i I such
More informationAdaptive Numerical Solution of State Constrained Optimal Control Problems
Technische Universität München Lehrstuhl für Mathematische Optimierung Adaptive Numerical Solution of State Constrained Optimal Control Problems Olaf Benedix Vollständiger Abdruck der von der Fakultät
More informationDefinition: Group A group is a set G together with a binary operation on G, satisfying the following axioms: a (b c) = (a b) c.
Algebraic Structures Abstract algebra is the study of algebraic structures. Such a structure consists of a set together with one or more binary operations, which are required to satisfy certain axioms.
More informationMATH REVIEW KIT. Reproduced with permission of the Certified General Accountant Association of Canada.
MATH REVIEW KIT Reproduced with permission of the Certified General Accountant Association of Canada. Copyright 00 by the Certified General Accountant Association of Canada and the UBC Real Estate Division.
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More information2 FirstOrder Equations: Method of Characteristics
2 FirstOrder Equations: Method of Characteristics In this section, we describe a general technique for solving firstorder equations. We begin with linear equations and work our way through the semilinear,
More informationSolvability of Laplace s Equation Kurt Bryan MA 436
1 Introduction Solvability of Laplace s Equation Kurt Bryan MA 436 Let D be a bounded region in lr n, with x = (x 1,..., x n ). We see a function u(x) which satisfies u = in D, (1) u n u = h on D (2) OR
More informationFIELDSMITACS Conference. on the Mathematics of Medical Imaging. Photoacoustic and Thermoacoustic Tomography with a variable sound speed
FIELDSMITACS Conference on the Mathematics of Medical Imaging Photoacoustic and Thermoacoustic Tomography with a variable sound speed Gunther Uhlmann UC Irvine & University of Washington Toronto, Canada,
More informationA FIRST COURSE IN OPTIMIZATION THEORY
A FIRST COURSE IN OPTIMIZATION THEORY RANGARAJAN K. SUNDARAM New York University CAMBRIDGE UNIVERSITY PRESS Contents Preface Acknowledgements page xiii xvii 1 Mathematical Preliminaries 1 1.1 Notation
More informationMarkov Chains, Stochastic Processes, and Advanced Matrix Decomposition
Markov Chains, Stochastic Processes, and Advanced Matrix Decomposition Jack Gilbert Copyright (c) 2014 Jack Gilbert. Permission is granted to copy, distribute and/or modify this document under the terms
More informationMathematics Notes for Class 12 chapter 12. Linear Programming
1 P a g e Mathematics Notes for Class 12 chapter 12. Linear Programming Linear Programming It is an important optimization (maximization or minimization) technique used in decision making is business and
More informationLet H and J be as in the above lemma. The result of the lemma shows that the integral
Let and be as in the above lemma. The result of the lemma shows that the integral ( f(x, y)dy) dx is well defined; we denote it by f(x, y)dydx. By symmetry, also the integral ( f(x, y)dx) dy is well defined;
More informationGalerkin Approximations and Finite Element Methods
Galerkin Approximations and Finite Element Methods Ricardo G. Durán 1 1 Departamento de Matemática, Facultad de Ciencias Exactas, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina. Chapter 1 Galerkin
More informationDISTRIBUTIONS AND FOURIER TRANSFORM
DISTRIBUTIONS AND FOURIER TRANSFORM MIKKO SALO Introduction. The theory of distributions, or generalized functions, provides a unified framework for performing standard calculus operations on nonsmooth
More informationTricyclic biregular graphs whose energy exceeds the number of vertices
MATHEMATICAL COMMUNICATIONS 213 Math. Commun., Vol. 15, No. 1, pp. 213222 (2010) Tricyclic biregular graphs whose energy exceeds the number of vertices Snježana Majstorović 1,, Ivan Gutman 2 and Antoaneta
More informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
More informationEXISTENCE AND NONEXISTENCE RESULTS FOR A NONLINEAR HEAT EQUATION
Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 5 (7), pp. 5 65. ISSN: 7669. UL: http://ejde.math.txstate.edu
More informationWAVES AND FIELDS IN INHOMOGENEOUS MEDIA
WAVES AND FIELDS IN INHOMOGENEOUS MEDIA WENG CHO CHEW UNIVERSITY OF ILLINOIS URBANACHAMPAIGN IEEE PRESS Series on Electromagnetic Waves Donald G. Dudley, Series Editor IEEE Antennas and Propagation Society,
More informationand s n (x) f(x) for all x and s.t. s n is measurable if f is. REAL ANALYSIS Measures. A (positive) measure on a measurable space
RAL ANALYSIS A survey of MA 641643, UAB 19992000 M. Griesemer Throughout these notes m denotes Lebesgue measure. 1. Abstract Integration σalgebras. A σalgebra in X is a nonempty collection of subsets
More information24. The Branch and Bound Method
24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NPcomplete. Then one can conclude according to the present state of science that no
More informationTowards a Theory for Arbitrarily Shaped Sound Field Reproduction Systems
Towards a Theory for Arbitrarily Shaped Sound Field Reproduction Systems Sascha Spors, Matthias Rath and Jens Ahrens Sascha.Spors@telekom.de Acoustics 08, Paris ===!" Laboratories AG Laboratories Aim of
More informationCourse 221: Analysis Academic year , First Semester
Course 221: Analysis Academic year 200708, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................
More informationADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB. Sohail A. Dianat. Rochester Institute of Technology, New York, U.S.A. Eli S.
ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB Sohail A. Dianat Rochester Institute of Technology, New York, U.S.A. Eli S. Saber Rochester Institute of Technology, New York, U.S.A. (g) CRC Press Taylor
More informationFINITE DIFFERENCE METHODS
FINITE DIFFERENCE METHODS LONG CHEN Te best known metods, finite difference, consists of replacing eac derivative by a difference quotient in te classic formulation. It is simple to code and economic to
More informationALMOST COMMON PRIORS 1. INTRODUCTION
ALMOST COMMON PRIORS ZIV HELLMAN ABSTRACT. What happens when priors are not common? We introduce a measure for how far a type space is from having a common prior, which we term prior distance. If a type
More information