**Graduate Studies in Mathematics**

Volume: 12;
1996;
164 pp;
Hardcover

MSC: Primary 35;

Print ISBN: 978-0-8218-0569-5

Product Code: GSM/12

List Price: $38.00

Individual Member Price: $30.40

**Electronic ISBN: 978-1-4704-2070-3
Product Code: GSM/12.E**

List Price: $38.00

Individual Member Price: $30.40

# Lectures on Elliptic and Parabolic Equations in Hölder Spaces

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*N. V. Krylov*

These lectures concentrate on fundamentals of the modern theory of linear elliptic and parabolic equations in Hölder spaces. Krylov shows that this theory—including some issues of the theory of nonlinear equations—is based on some general and extremely powerful ideas and some simple computations.

The main object of study is the first boundary-value problems for elliptic and parabolic equations, with some guidelines concerning other boundary-value problems such as the Neumann or oblique derivative problems or problems involving higher-order elliptic operators acting on the boundary. Numerical approximations are also discussed.

This book, with nearly 200 exercises, will provide a good understanding of what kinds of results are available and what kinds of techniques are used to obtain them.

#### Table of Contents

# Table of Contents

## Lectures on Elliptic and Parabolic Equations in Holder Spaces

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Preface ix10 free
- Chapter 1. Elliptic Equations with Constant Coefficients in R[sup(d)] 114 free
- 1.1. The notion of elliptic operator 114
- 1.2. Solvability. Green's representation 417
- 1.3. Green's functions as limits of usual functions 518
- 1.4. Green's functions as usual functions 720
- 1.5. Differentiability of Green's functions 821
- 1.6. Some properties of solutions of Lu = f 1023
- 1.7. Some further information on G 1225
- 1.8. Hints to exercises 1427

- Chapter 2. Laplace's Equation 1528
- 2.1. Green's identities 1528
- 2.2. The Poisson formula 1629
- 2.3. Green's functions in domains 1831
- 2.4. The Green's function and the Poisson kernel in a ball 2033
- 2.5. Some properties of harmonic functions 2134
- 2.6. The maximum principle 2336
- 2.7. Poisson's equation in a ball 2437
- 2.8. Other second-order elliptic operators with constant coefficients 2740
- 2.9. The maximum principle for second-order equations with variable coefficients 2942
- 2.10. Hints to exercises 3144

- Chapter 3. Solvability of Elliptic Equations with Constant Coefficients in the Holder Spaces 3346
- 3.1. The HÖlder spaces 3346
- 3.2. Interpolation inequalities 3548
- 3.3. Equivalent norms in the Holder spaces 3851
- 3.4. A priori estimates in the whole space for Laplace's operator 4053
- 3.5. An estimate for derivatives of L-harmonic functions 4255
- 3.6. A priori estimates in the whole space for general elliptic operators 4558
- 3.7. Solvability of elliptic equations with constant coefficients 4760
- 3.8. Hints to exercises 4962

- Chapter 4. Elliptic Equations with Variable Coefficients in R[sup(d)] 5164
- 4.1. Schauder's a priori estimates 5164
- 4.2. Better regularity of Lu implies better regularity of u 5467
- 4.3. Solvability of second-order elliptic equations with variable coefficients. The method of continuity 5568
- 4.4. The case of second-order equations Lu — zu = f with z complex 5972
- 4.5. Solvability of higher-order elliptic equations with variable coefficients 6174
- 4.6. Hints to exercises 6376

- Chapter 5. Second-Order Elliptic Equations in Half Spaces 6578
- 5.1. More equivalent norms in the Holder spaces 6679
- 5.2. Laplace's equation in half spaces 6780
- 5.3. Poisson's equation in half spaces 6982
- 5.4. Solvability of elliptic equations with variable coefficients in half spaces 7184
- 5.5. Remarks on the Neumann and other boundary-value problems in half spaces 7386
- 5.6. Hints to exercises 7689

- Chapter 6. Second-Order Elliptic Equations in Smooth Domains 7790
- 6.1. The maximum principle. Domains of class C[sup(r)] 7790
- 6.2. Equations near the boundary 7992
- 6.3. Partitions of unity and a priori estimates 8194
- 6.4. The regularizer 8396
- 6.5. The existence theorems 8497
- 6.6. Finite-difference approximations of elliptic operators 8598
- 6.7. Convergence of numerical approximations 87100
- 6.8. Hints to exercises 88101

- Chapter 7. Elliptic Equations in Non-Smooth Domains 91104
- 7.1. Interior a priori estimates 91104
- 7.2. Generalized solutions of the Dirichlet problem with zero boundary condition 93106
- 7.3. Generalized solutions of the Dirichlet problem with continuous boundary conditions 96109
- 7.4. Some properties of generalized solutions 97110
- 7.5. An example 100113
- 7.6. Barriers and the exterior cone condition 101114
- 7.7. Hints to exercises 104117

- Chapter 8. Parabolic Equations in the Whole Space 105118
- 8.1. The maximum principle 105118
- 8.2. The Cauchy problem, semigroup approach, motivation 110123
- 8.3. Proof of Theorem 8.2.1 112125
- 8.4. The heat equation 115128
- 8.5. Parabolic Holder spaces 117130
- 8.6. The basic a priori estimate 121134
- 8.7. Solvability of the heat equation in the Holder spaces 122135
- 8.8. Parabolic interpolation inequalities 124137
- 8.9. The Schauder a priori estimates 127140
- 8.10. The existence theorems 129142
- 8.11. Interior a priori estimates 130143
- 8.12. Better regularity of solutions 131144
- 8.13. Hints to exercises 133146

- Chapter 9. Boundary-Value Problems for Parabolic Equations in Half Spaces 137150
- Chapter 10. Parabolic Equations in Domains 147160
- Bibliography 161174
- Index 163176 free
- Back Cover 165178

#### Readership

Graduate students and researchers in mathematics, physics, and engineering interested in the theory of partial differential equations.

#### Reviews

Short but not condensed, well organized and gives a stimulating presentation of basic aspects of the theory of elliptic and parabolic equations in Hölder spaces … an interesting addition for students and instructors.

-- Zentralblatt MATH

The author has fully achieved his goal … and has written an impressive book that presents nice material in an interesting way … this book can be recommended as a thorough, modern and sufficiently broad introduction to partial differential equations of elliptic and parabolic types for graduate students and instructors (and also for individual study) in mathematics, physics, and (possibly) engineering.

-- Mathematical Reviews