Complex Made Simple (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 97)

perhaps Uniquely Among Mathematical Topics, Complex Analysis Presents The Student With The Opportunity To Learn A Thoroughly Developed Subject That Is Rich In Both Theory And Applications. Even In An Introductory Course, The Theorems And Techniques Can Have Elegant Formulations. But For Any Of These Profound Results, The Student Is Often Left Asking: What Does It Really Mean? Where Does It Come From? In Complex Made Simple, David Ullrich Shows The Student How To Think Like An Analyst. In Many Cases, Results Are Discovered Or Derived, With An Explanation Of How The Students Might Have Found The Theorem On Their Own. Ullrich Explains Why A Proof Works. He Will Also, Sometimes, Explain Why A Tempting Idea Does Not Work. Complex Made Simple Looks At The Dirichlet Problem For Harmonic Functions Twice: Once Using The Poisson Integral For The Unit Disk And Again In An Informal Section On Brownian Motion, Where The Reader Can Understand Intuitively How The Dirichlet Problem Works For General Domains. Ullrich Also Takes Considerable Care To Discuss The Modular Group, Modular Function, And Covering Maps, Which Become Important Ingredients In His Modern Treatment Of The Often-overlooked Original Proof Of The Big Picard Theorem. This Book Is Suitable For A First-year Course In Complex Analysis. The Exposition Is Aimed Directly At The Students, With Plenty Of Details Included. The Prerequisite Is A Good Course In Advanced Calculus Or Undergraduate Analysis. This Book Concentrates On The Basic Facts And Ideas Of The Modern Theory Of Linear Elliptic And Parabolic Equations In Sobolev Spaces. The Main Areas Covered In This Book Are The First Boundary-value Problem For Elliptic Equations And The Cauchy Problem For Parabolic Equations. In Addition, Other Boundary-value Problems Such As The Neumann Or Oblique Derivative Problems Are Briefly Covered. As Is Natural For A Textbook, The Main Emphasis Is On Organizing Well-known Ideas In A Self-contained Exposition. Among The Topics Included That Are Not Usually Covered In A Textbook Are A Relatively Recent Development Concerning Equations With Vmo Coefficients And The Study Of Parabolic Equations With Coefficients Measurable Only With Respect To The Time Variable. There Are Numerous Exercises Which Help The Reader Better Understand The Material. After Going Through The Book, The Reader Will Have A Good Understanding Of Results Available In The Modern Theory Of Partial Differential Equations And The Technique Used To Obtain Them. Prerequisites Are Basics Of Measure Theory, The Theory Of L[subscript P] Spaces, And The Fourier Transform.--jacket. Chapter 1. Second-order Elliptic Equations In $w^{2}_{2}(\mathbb {r}^{d}) Tchapter 2. Second-order Parabolic Equations In $w^{1,k}_{2}(\mathbb {r}^{d+1}) Tchapter 3. Some Tools From Real Analysis Chapter 4. Basic $\mathcal {l}_{p}$-estimates For Parabolic And Elliptic Equations Chapter 5. Parabolic And Elliptic Equations In $w^{1,k}_{p}$ And $w^{k}_{p} Tchapter 6. Equations With Vmo Coefficients Chapter 7. Parabolic Equations With Vmo Coefficients In Spaces With Mixed Norms Chapter 8. Second-order Elliptic Equations In $w^{2}_{p}(\omega ) Tchapter 9. Second-order Elliptic Equations In $w^{k}_{p}(\omega ) Tchapter 10. Sobolev Embedding Theorems For $w^{k}_{p}(\omega ) Tchapter 11. Second-order Elliptic Equations $lu-\lambda U=f$ With $\lambda $ Small Chapter 12. Fourier Transform And Elliptic Operators Chapter 13. Elliptic Operators And The Spaces $h^{\gamma }_{p}$ N.v. Krylov. Includes Bibliographical References (p. 353-354) And Index. This book concentrates on the basic facts and ideas of the modern theory of linear elliptic and parabolic equations in Sobolev spaces. The main areas covered in this book are the first boundary-value problem for elliptic equations and the Cauchy problem for parabolic equations. In addition, other boundary-value problems such as the Neumann or oblique derivative problems are briefly covered. As is natural for a textbook, the main emphasis is on organizing well-known ideas in a self-contained exposition. Among the topics included that are not usually covered in a textbook are a relatively recent development concerning equations with $\mathsf{VMO}$ coefficients and the study of parabolic equations with coefficients measurable only with respect to the time variable. There are numerous exercises which help the reader better understand the material. After going through the book, the reader will have a good understanding of results available in the modern theory of partial differential equations and the technique used to obtain them. Prerequisites are basics of measure theory, the theory of $L_p$ spaces, and the Fourier transform. This book concentrates on the basic facts and ideas of the modern theory of linear elliptic and parabolic equations in Sobolev spaces. The main areas covered in this book are the first boundary-value problem for elliptic equations and the Cauchy problem for parabolic equations. In addition, other boundary-value problems such as the Neumann or oblique derivative problems are briefly covered. As is natural for a textbook, the main emphasis is on organizing well-known ideas in a self-contained exposition. Among the topics included that are not usually covered in a textbook are a relatively recent development concerning equations with $\textsf{VMO}$ coefficients and the study of parabolic equations with coefficients measurable only with respect to the time variable. There are numerous exercises which help the reader better understand the material. After going through the book, the reader will have a good understanding of results available in the modern theory of partial differential equations and the technique used to obtain them. Prerequisites are basics of measure theory, the theory of $L_p$ spaces, and the Fourier transform.