Lectures on Elliptic and Parabolic Equations in Sobolev Spaces

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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.

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Product Details

  • ISBN-13: 9780821846841
  • Publisher: American Mathematical Society
  • Publication date: 9/19/2008
  • Series: Graduate Studies in Mathematics Series, #96
  • Pages: 357
  • Product dimensions: 7.30 (w) x 10.20 (h) x 1.00 (d)

Table of Contents

Ch. 1 Second-order elliptic equations in W[subscript 2][superscript 2](R[superscript d]) 1

Ch. 2 Second-order parabolic equations in W[subscript 2][superscript 1,k](R[superscript d+1]) 45

Ch. 3 Some tools from real analysis 73

Ch. 4 Basic L[subscript p] estimates for parabolic and elliptic equations 93

Ch. 5 Parabolic and elliptic equations in W[subscript p][superscript 1,k] and W[subscript p][superscript k] 117

Ch. 6 Equations with VMO coefficients 125

Ch. 7 Parabolic equations with VMO coefficients in spaces with mixed norms 145

Ch. 8 Second-order elliptic equations in W[subscript p][superscript 2]([Omega]) 157

Ch. 9 Second-order elliptic equations in W[subscript p][superscript k]([Omega]) 181

Ch. 10 Sobolev embedding theorems for W[subscript p][superscript k]([Omega]) 201

Ch. 11 Second-order elliptic equations Lu - [lambda]u = f with [lambda] small 231

Ch. 12 Fourier transform and elliptic operators 267

Ch. 13 Elliptic operators and the spaces H[subscript p][superscript [gamma]] 311

Bibliography 353

Index 355

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