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Book DetailsGraduate Studies in MathematicsVolume: 96; 2008; 357 ppMSC: Primary 35
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 boundaryvalue problem for elliptic equations and the Cauchy problem for parabolic equations. In addition, other boundaryvalue problems such as the Neumann or oblique derivative problems are briefly covered. As is natural for a textbook, the main emphasis is on organizing wellknown ideas in a selfcontained 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.
ReadershipGraduate students and research mathematicians interested in partial differential equations.

Table of Contents

Chapters

Chapter 1. Secondorder elliptic equations in $W^{2}_{2}(\mathbb {R}^{d})$

Chapter 2. Secondorder parabolic equations in $W^{1,k}_{2}(\mathbb {R}^{d+1})$

Chapter 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 $W^{k}_{p}$

Chapter 6. Equations with VMO coefficients

Chapter 7. Parabolic equations with VMO coefficients in spaces with mixed norms

Chapter 8. Secondorder elliptic equations in $W^{2}_{p}(\Omega )$

Chapter 9. Secondorder elliptic equations in $W^{k}_{p}(\Omega )$

Chapter 10. Sobolev embedding theorems for $W^{k}_{p}(\Omega )$

Chapter 11. Secondorder elliptic equations $Lu\lambda u=f$ with $\lambda $ small

Chapter 12. Fourier transform and elliptic operators

Chapter 13. Elliptic operators and the spaces $H^{\gamma }_{p}$


Additional Material

Reviews

This book is certain to become a source of inspiration for every researcher in nonlinear analysis. [The book] is beautifully written and well organized, and I strongly recommend this book to anyone seeking a stylish, balanced, uptodate survey of this central area of the modern nonlinear analysis.
Mathematical Reviews


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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 boundaryvalue problem for elliptic equations and the Cauchy problem for parabolic equations. In addition, other boundaryvalue problems such as the Neumann or oblique derivative problems are briefly covered. As is natural for a textbook, the main emphasis is on organizing wellknown ideas in a selfcontained 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.
Graduate students and research mathematicians interested in partial differential equations.

Chapters

Chapter 1. Secondorder elliptic equations in $W^{2}_{2}(\mathbb {R}^{d})$

Chapter 2. Secondorder parabolic equations in $W^{1,k}_{2}(\mathbb {R}^{d+1})$

Chapter 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 $W^{k}_{p}$

Chapter 6. Equations with VMO coefficients

Chapter 7. Parabolic equations with VMO coefficients in spaces with mixed norms

Chapter 8. Secondorder elliptic equations in $W^{2}_{p}(\Omega )$

Chapter 9. Secondorder elliptic equations in $W^{k}_{p}(\Omega )$

Chapter 10. Sobolev embedding theorems for $W^{k}_{p}(\Omega )$

Chapter 11. Secondorder elliptic equations $Lu\lambda u=f$ with $\lambda $ small

Chapter 12. Fourier transform and elliptic operators

Chapter 13. Elliptic operators and the spaces $H^{\gamma }_{p}$

This book is certain to become a source of inspiration for every researcher in nonlinear analysis. [The book] is beautifully written and well organized, and I strongly recommend this book to anyone seeking a stylish, balanced, uptodate survey of this central area of the modern nonlinear analysis.
Mathematical Reviews