Lectures on Elliptic and Parabolic Equations in Sobolev Spaces (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 96)

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. 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. Focuses on the basic facts and ideas of the modern theory of linear elliptic and parabolic equations in Sobolev spaces. This book covers the first boundary-value problem for elliptic equations and the Cauchy problem for parabolic equations. It also covers other boundary-value problems such as the Neumann or oblique derivative problems.