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E-raamat: Semigroups of Bounded Operators and Second-Order Elliptic and Parabolic Partial Differential Equations

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(University of Salerno), (University of Parma)
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Semigroups of Bounded Operators and Second-Order Elliptic and Parabolic Partial Differential EquationsSuurem pilt
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"Semigroups of Bounded Operators and Second-Order Elliptic and Parabolic Partial Differential Equations aims to propose a unified approach to elliptic and parabolic equations with bounded and smooth coefficients. The book will highlight the connections between these equations and the theory of semigroups of operators, while demonstrating how the theory of semigroups represents a powerful tool to analyze general parabolic equations. Features Useful for students and researchers as an introduction to the field of partial differential equations of elliptic and parabolic type Introduces the reader to the theory of operator semigroups as a tool for the analysis of partial differential equations"--

Semigroups of Bounded Operators and Second-Order Elliptic and Parabolic Partial Differential Equations aims to propose a unified approach to elliptic and parabolic equations with bounded and smooth coefficients. The book will highlight the connections between these equations and the theory of semigroups of operators, while demonstrating how the theory of semigroups represents a powerful tool to analyze general parabolic equations.

Features

  • Useful for students and researchers as an introduction to the field of partial differential equations of elliptic and parabolic type

    • Introduces the reader to the theory of operator semigroups as a tool for the analysis of partial differential equations.

    • Symbol Description xi
    Preface xiii
    Introduction xv
    1 Function Spaces
    		1(34)
    1.1 Spaces of (Holder) Continuous Functions
    		1(7)
    1.1.1 Functions defined on the boundary of a smooth open set
    		7(1)
    1.2 Anisotropic and Parabolic Spaces of Holder Continuous Functions
    		8(7)
    1.2.1 Anisotropic spaces of functions defined on the boundary of a set
    		14(1)
    1.3 Lp- and Sobolev Spaces
    		15(11)
    1.4 Besov Spaces
    		26(8)
    1.5 Exercises
    		34(1)
    I Semigroups of Bounded Operators
    		35(48)
    2 Strongly Continuous Semigroups
    		37(18)
    2.1 Definitions and Basic Properties
    		38(2)
    2.2 The Infinitesimal Generator
    		40(4)
    2.3 The Hille-Yosida, Lumer-Phillips and Trotter-Kato Theorems
    		44(6)
    2.4 Nonhomogeneous Cauchy Problems
    		50(3)
    2.5 Notes and Remarks
    		53(1)
    2.6 Exercises
    		54(1)
    3 Analytic Semigroups
    		55(28)
    3.1 Prelude
    		56(1)
    3.2 Sectorial Operators and Analytic Semigroups
    		57(8)
    3.3 Interpolation Spaces
    		65(5)
    3.4 Nonhomogeneous Cauchy Problems
    		70(9)
    3.5 Notes
    		79(1)
    3.6 Exercises
    		80(3)
    II Parabolic Equations
    		83(164)
    4 Elliptic and Parabolic Maximum Principles
    		85(22)
    4.1 The Parabolic Maximum Principles
    		85(12)
    4.1.1 Parabolic weak maximum principle
    		85(7)
    4.1.2 The strong maximum principle
    		92(5)
    4.2 Elliptic Maximum Principles
    		97(7)
    4.3 Notes
    		104(1)
    4.4 Exercises
    		104(3)
    5 Prelude to Parabolic Equations: The Heat Equation and the Gauss-Weierstrass Semigroup in Ct(Rd)
    		107(30)
    5.1 The Homogeneous Heat Equation in Rd. Classical Solutions: Existence and Uniqueness
    		108(3)
    5.2 The Gauss-Weierstrass Semigroup
    		111(9)
    5.2.1 Estimates of the spatial derivatives of T(t)f
    		114(6)
    5.3 Two Equivalent Characterizations of Holder Spaces
    		120(5)
    5.4 Optimal Schauder Estimates
    		125(11)
    5.5 Notes
    		136(1)
    5.6 Exercises
    		136(1)
    6 Parabolic Equations in IRd
    		137(38)
    6.1 The Continuity Method
    		138(1)
    6.2 A priori Estimates
    		139(18)
    6.2.1 Solving problem (6.0.1)
    		145(1)
    6.2.2 Interior Schauder estimates for solutions to parabolic equations in domains: Part I
    		146(11)
    6.3 More on the Cauchy Problem (6.0.1)
    		157(6)
    6.4 The Semigroup Associated with the Operator A
    		163(8)
    6.4.1 Interior Schauder estimates for solutions to parabolic equations in domains: Part II
    		170(1)
    6.5 Higher-Order Regularity Results
    		171(3)
    6.6 Notes
    		174(1)
    6.7 Exercises
    		174(1)
    7 Parabolic Equations in with Dirichlet Boundary Conditions
    		175(24)
    7.1 Technical Results
    		176(8)
    7.2 An Auxiliary Boundary Value Problem
    		184(6)
    7.3 Proof of Theorem 7.0.2 and a Corollary
    		190(2)
    7.4 More on the Cauchy Problem (7.0.1)
    		192(1)
    7.5 The Associated Semigroup
    		193(4)
    7.6 Notes
    		197(1)
    7.7 Exercises
    		197(2)
    8 Parabolic Equations in IR+ with More General Boundary Conditions
    		199(30)
    8.1 A Priori Estimates
    		200(9)
    8.2 Proof of Theorem 8.0.2
    		209(10)
    8.3 Interior Schauder Estimates for Solutions to Parabolic Equations in Domains: Part III
    		219(5)
    8.4 More on the Cauchy Problem (8.0.1)
    		224(1)
    8.5 The Associated Semigroup
    		225(3)
    8.6 Exercises
    		228(1)
    9 Parabolic Equations in Bounded Smooth Domains 0
    		229(18)
    9.1 Optimal Schauder Estimates for Solutions to Problems (9.0.1) and (9.0.2)
    		230(10)
    9.2 Interior Schauder Estimates for Solutions to Parabolic equations in Domains: Part IV
    		240(3)
    9.3 More on the Cauchy Problems (9.0.1) and (9.0.2)
    		243(2)
    9.4 The Associated Semigroup
    		245(1)
    9.5 Exercises
    		246(1)
    III Elliptic Equations
    		247(168)
    10 Elliptic Equations in Rd
    		249(42)
    10.1 Solutions in Holder Spaces
    		251(5)
    10.1.1 The Laplace equation
    		251(1)
    10.1.2 More general elliptic operators
    		252(2)
    10.1.3 Further regularity results and interior estimates
    		254(2)
    10.2 Solutions in Lp (Rd;C) (p ε (1,∞)
    		256(30)
    10.2.1 The Calderon-Zygmund inequality
    		256(10)
    10.2.2 The Laplace equation
    		266(3)
    10.2.3 More general elliptic operators
    		269(10)
    10.2.4 Further regularity results and interior Lp-estimates
    		279(7)
    10.3 Solutions in L∞(Rd;C) and in Cb(Rd;C)
    		286(4)
    10.4 Exercises
    		290(1)
    11 Elliptic Equations in Rd with Homogeneous Dirichlet Boundary Conditions
    		291(24)
    11.1 Solutions in Holder Spaces
    		292(4)
    11.1.1 Further regularity results
    		294(2)
    11.2 Solutions in Sobolev Spaces
    		296(13)
    11.2.1 Further regularity results
    		304(5)
    11.3 Solutions in L∞(Rd+;C) and in Cb(Rd+;C)
    		309(5)
    11.4 Exercises
    		314(1)
    12 Elliptic Equations in R+ with General Boundary Conditions
    		315(24)
    12.1 TheCa-Theory
    		316(5)
    12.1.1 Further regularity
    		319(2)
    12.2 Elliptic Equations in Lp(Rd+;C)
    		321(13)
    12.2.1 Further regularity results
    		332(2)
    12.3 Solutions in L∞(Rd+ ;C) and in Cb(Rd+;C)
    		334(4)
    12.4 Exercises
    		338(1)
    13 Elliptic Equations on Smooth Domains ft
    		339(30)
    13.1 Elliptic Equations in Cα(ω;C)
    		339(12)
    13.1.1 Further regularity results
    		348(3)
    13.2 Elliptic Equations in Lp (ω;C)
    		351(13)
    13.2.1 Further regularity results
    		360(4)
    13.3 Solutions in L∞(ω; C), in C(ω; C) and in Cb(ω; C)
    		364(4)
    13.4 Exercises
    		368(1)
    14 Elliptic Operators and Analytic Semigroups
    		369(28)
    14.1 The Semigroup in Cb(Rd;C)
    		369(5)
    14.2 The Semigroups in Cb(Rd+;C)
    		374(15)
    14.2.1 Proof of Theorems 7.4.1 and 7.4.3
    		384(5)
    14.3 The Semigroups in Cb(ω; C)
    		389(6)
    14.4 Exercises
    		395(2)
    15 Kernel Estimates
    		397(18)
    15.1 Dunford-Pettis Criterion and Ultracontractivity
    		397(3)
    15.2 Gaussian Estimates for Second-Order Elliptic Operators with Dirichlet Boundary Conditions
    		400(12)
    15.3 Integral Representation for the Semigroups in
    Chapters 6, 7 and 9
    		412(1)
    15.4 Notes
    		413(1)
    15.5 Exercises
    		414(1)
    IV Appendices
    		415(2)
    A Basic Notions of Functional Analysis in Banach Spaces
    		417(58)
    A.1 Bounded and Closed Linear Operators
    		417(1)
    A.2 Vector Valued Riemann Integral
    		418(4)
    A.3 Holomorphic functions
    		422(2)
    A.4 Spectrum and Resolvent
    		424(2)
    A.5 A Few Basic Notions from Interpolation Theory
    		426(14)
    A.5.1 Marcinkiewicz's Interpolation Theorem
    		437(3)
    A.6 Exercises
    		440(1)
    B Smooth Domains and Extension Operators
    		441(1)
    B.1 Partition of Unity
    		441(2)
    B.2 Smooth Domains
    		443(7)
    B.3 Traces of Functions in Sobolev Spaces
    		450(4)
    B.4 Extension Operators
    		454(21)
    B.4.1 Extending functions defined on open sets
    		454(6)
    B.4.2 Extending functions defined on the boundary of a set
    		460(15)
    Bibliography 475(4)
    Index 479
    Luca Lorenzi is Full Professor of Mathematical Analysis at University of Parma (Italy). He received his PhD in Mathematics from the University of Pisa (Italy) in 2001. In 2000 he got a permanent position as assistant professor in Mathematical Analysis at the University of Parma and he was promoted to associate professor in 2006 still at the University of Parma. In 2013 he got the Italian National habilitation as full professor in Mathematical Analysis. His research interests are mainly focused on Partial Differential Equations. In particular, Evolution Equations and Operator Semigroups, Non-autonomous Differential Equations, Elliptic and Parabolic Differential Operators with Unbounded Coefficients. He has authored or co-authored over 60 papers published on international journals and three scientific monographs.

    Abdelaziz Rhandi is Full Professor of Mathematical Analysis at University of Salerno (Italy). Before joining Salerno University in 2006, he served as Full Professor of Mathematics at the University of Marrakesh (Morocco) since 1999. He received his first PhD in Mathematics from the University of Besançon (France) and the second one from the University of Tübingen (Germany). In 2003, he got the Alexander von Humboldt Fellow and was the winner of the 2006 HP Technology for Teaching Higher Education Prize. He has worked as a visiting professor in several universities in Algeria, France, Germany, Italy, Tunisia and USA. His research interests are mainly focused on Applied Functional Analysis and Partial Differential Equations. In particular, Evolution Equations and Operator Semigroups, Non-autonomous Differential Equations, Elliptic and Parabolic Differential Operators with Unbounded Coefficients.
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