Lectures on Elliptic and Parabolic Equations in Sobolev Spaces

by Krylov, N. V.

• ISBN13: 9780821846841
• ISBN10: 0821846841
• Format: Hardcover
• Copyright: 2008-09-19
• Publisher: Amer Mathematical Society

Summary

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 equationswith $\textsf{VMO}$ 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. Prerequesites are basics of measure theory, the theory of $L p$ spaces, and the Fourier transform.

Table of Contents

Second-order elliptic equations in $W^{2}_{2}(\mathbb{R}^{d})$

Second-order parabolic equations in $W^{1,k}_{2}(\mathbb{R}^{d+1})$

Some tools from real analysis Basic $\mathcal{L}_{p}$-estimates for parabolic and elliptic equations

Parabolic and elliptic equations in $W^{1,k}_{p}$ and $W^{k}_{p}$

Equations with VMO coefficients Parabolic equations with VMO coefficients in spaces with mixed norms

Second-order elliptic equations in $W^{2}_{p}(\Omega)$

Second-order elliptic equations in $W^{k}_{p}(\Omega)$

Sobolev embedding theorems for $W^{k}_{p}(\Omega)$

Second-order elliptic equations $Lu-\lambda u=f$ with $\lambda$ small

Fourier transform and elliptic operators

Elliptic operators and the spaces $H^{\gamma}_{p}$

Bibliography

Index

Table of Contents provided by Publisher. All Rights Reserved.