**Graduate Studies in Mathematics**

Volume: 96;
2008;
357 pp;
Hardcover

MSC: Primary 35;

Print ISBN: 978-0-8218-4684-1

Product Code: GSM/96

List Price: $71.00

Individual Member Price: $56.80

**Electronic ISBN: 978-1-4704-2121-2
Product Code: GSM/96.E**

List Price: $71.00

Individual Member Price: $56.80

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#### Supplemental Materials

# Lectures on Elliptic and Parabolic Equations in Sobolev Spaces

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*N. V. Krylov*

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.

#### Table of Contents

# Table of Contents

## Lectures on Elliptic and Parabolic Equations in Sobolev Spaces

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Preface xi12 free
- Chapter 1. Second-order elliptic equations in W[sup(2)][sub(2)](R[sup(d)]) 120 free
- §1. The simplest equation λu „ Δu = f 221
- §2. Integrating the determinants of Hessians (optional) 726
- §3. Sobolev spaces W[sup(k)][sub(p)](Ω) 827
- §4. Second-order elliptic differential operators 1433
- §5. Multiplicative inequalities 1736
- §6. Solvability of elliptic equations with continuous coefficients 2039
- §7. Higher regularity of solutions 2544
- §8. Sobolev mollifiers 3251
- §9. Singular-integral representation of u[sub(xx)] 3857
- §10. Hints to exercises 4261

- Chapter 2. Second-order parabolic equations in W[sup(1,k)][sub(2)](R[sup(d+1)]) 4564
- Chapter 3. Some tools from real analysis 7392
- Chapter 4. Basic L[sub(p)] estimates for parabolic and elliptic equations 93112
- Chapter 5. Parabolic and elliptic equations in W[sup(1,k)][sub(p)] and W[sup(k)][sub(p)] 117136
- Chapter 6. Equations with VMO coefficients 125144
- §1. Estimating L[sub(q)] oscillations of u[sub(xx)] 125144
- §2. Estimating sharp functions of u[sub(xx)] 130149
- §3. A priori estimates for parabolic and elliptic equations with VMO coefficients 134153
- §4. Solvability of parabolic and elliptic equations with VMO coefficients. The Cauchy problem 139158
- §5. Hints to exercises 143162

- Chapter 7. Parabolic equations with VMO coefficients in spaces with mixed norms 145164
- Chapter 8. Second-order elliptic equations in W[sup(2)][sub(p)](Ω) 157176
- Chapter 9. Second-order elliptic equations in W[sup(k)][sub(p)](Ω) 181200
- Chapter 10. Sobolev embedding theorems for W[sup(k)][sub(p)](Ω) 201220
- §1. Embedding for Campanato and Slobodetskii spaces 203222
- §2. Embedding W[sup(1)][sub(p)](Ω) ⊂ C[sup(1-d/p)](Ω). Morrey's theorem 209228
- §3. The Gagliardo-Nirenberg theorem 215234
- §4. General embedding theorems 216235
- §5. Compactness of embeddings. Kondrashov's theorem 223242
- §6. An application of Riesz's theory of compact operators 227246
- §7. Hints to exercises 229248

- Chapter 11. Second-order elliptic equations Lu „ λu = f with λ small 231250
- §1. Maximum principle for smooth functions 232251
- §2. Resolvent operator for λlarge 236255
- §3. Solvability of equations in smooth domains 241260
- §4. The way we proceed further 245264
- §5. Decay at infinity of solutions of Lu = f in R[sup(d)] 246265
- §6. Equations in R[sup(d)] with λ small 249268
- §7. Traces of W[sup(k)][sub(p)](Ω) functions on dft 255274
- §8. The maximum principle in W[sup(2)][sub(p)]. Green's functions 262281
- §9. Hints to exercises 263282

- Chapter 12. Fourier transform and elliptic operators 267286
- §1. The space S 268287
- §2. The notion of elliptic differential operator 269288
- §3. Comments on the oblique derivative and other boundary-value problems. Instances of pseudo-differential operators 272291
- §4. Pseudo-differential operators 274293
- §5. Green's functions 280299
- §6. Existence of Green's functions 285304
- §7. Estimating G and its derivatives 288307
- §8. Boundedness of the zeroth-order pseudo-differential operators in L[sub(p)] 293312
- §9. Operators related to the Laplacian 297316
- §10. An embedding lemma 304323
- §11. Hints to exercises 308327

- Chapter 13. Elliptic operators and the spaces H[sup(γ)][sub(p)] 311330
- §1. The space H 311330
- §2. Some properties of the space H 315334
- §3. The spaces H[sup(γ)][sub(p)] 317336
- §4. Higher-order elliptic differential equations with continuous coefficients in H[sup(γ)][sub(p)] 328347
- §5. Second-order parabolic equations. Semigroups (optional) 333352
- §6. Second-order divergence type elliptic equations with continuous coefficients 335354
- §7. Nonzero Dirichlet condition and traces 339358
- §8. Sobolev embedding theorems for H[sup(γ)][sub(p)] spaces 342361
- §9. Sobolev mollifiers 346365
- §10. Hints to exercises 350369

- Bibliography 353372
- Index 355374 free
- Back Cover Back Cover1377

#### Readership

Graduate students and research mathematicians interested in partial differential equations.

#### 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, up-to-date survey of this central area of the modern nonlinear analysis.

-- Mathematical Reviews