Elliptic Partial Differential Equations of Second Order

by David Gilbarg; Neil S. Trudinger

• ISBN13: 9783540130253
• ISBN10: 354013025X
• Format: Nonspecific Binding
• Copyright: 2015-03-30
• Publisher: Springer Nature

Summary

From the reviews: "This is a book of interest to any having to work with differential equations, either as a reference or as a book to learn from. The authors have taken trouble to make the treatment self-contained. It (is) suitable required reading for a PhD student. Although the material has been developed from lectures at Stanford, it has developed into an almost systematic coverage that is much longer than could be covered in a year's lectures," Newsletter, New Zealand Mathematical Society, 1985 "Primarily addressed to graduate students this elegant book is accessible and useful to a broad spectrum of applied mathematicians," Revue Roumaine de Math??matiques Pures et Appliqu??es,1985

Table of Contents

Chapter 1. Introduction	1	(12)
Part I. Linear Equations	13	(246)
Chapter 2. Laplace's Equation	13	(18)
2.1. The Mean Value Inequalities	13	(2)
2.2. Maximum and Minimum Principle	15	(1)
2.3. The Harnack Inequality	16	(1)
2.4. Green's Representation	17	(2)
2.5. The Poisson Integral	19	(2)
2.6. Convergence Theorems	21	(1)
2.7. Interior Estimates of Derivatives	22	(1)
2.8. The Dirichlet Problem; the Method of Subharmonic Functions	23	(4)
2.9. Capacity	27	(1)
Problems	28	(3)
Chapter 3. The Classical Maximum Principle	31	(20)
3.1. The Weak Maximum Principle	32	(1)
3.2. The Strong Maximum Principle	33	(3)
3.3. Apriori Bounds	36	(1)
3.4. Gradient Estimates for Poisson's Equation	37	(4)
3.5. A Harnack Inequality	41	(4)
3.6. Operators in Divergence Form	45	(1)
Notes	46	(1)
Problems	47	(4)
Chapter 4. Poisson's Equation and the Newtonian Potential	51	(22)
4.1. Holder Continuity	51	(3)
4.2. The Dirichlet Problem for Poisson's Equation	54	(2)
4.3. Holder Estimates for the Second Derivatives	56	(8)
4.4. Estimates at the Boundary	64	(3)
4.5. Holder Estimates for the First Derivatives	67	(3)
Notes	70	(1)
Problems	70	(3)
Chapter 5. Banach and Hilbert Spaces	73	(14)
5.1. The Contraction Mapping Principle	74	(1)
5.2. The Method of Continuity	74	(1)
5.3. The Fredholm Alternative	75	(4)
5.4. Dual Spaces and Adjoints	79	(1)
5.5. Hilbert Spaces	80	(1)
5.6. The Projection Theorem	81	(1)
5.7. The Riesz Representation Theorem	82	(1)
5.8. The Lax-Milgram Theorem	83	(1)
5.9. The Fredholm Alternative in Hilbert Spaces	83	(2)
5.10. Weak Compactness	85	(1)
Notes	85	(1)
Problems	86	(1)
Chapter 6. Classical Solutions; the Schauder Approach	87	(57)
6.1. The Schauder Interior Estimates	89	(5)
6.2. Boundary and Global Estimates	94	(6)
6.3. The Dirichlet Problem	100	(9)
6.4. Interior and Boundary Regularity	109	(3)
6.5. An Alternative Approach	112	(4)
6.6. Non-Uniformly Elliptic Equations	116	(4)
6.7. Other Boundary Conditions; the Oblique Derivative Problem	120	(10)
6.8. Appendix 1: Interpolation Inequalities	130	(6)
6.9. Appendix 2: Extension Lemmas	136	(2)
Notes	138	(3)
Problems	141	(3)
Chapter 7. Sobolev Spaces	144	(33)
7.1. L^P Spaces	145	(2)
7.2. Regularization and Approximation by Smooth Functions	147	(2)
7.3. Weak Derivatives	149	(2)
7.4. The Chain Rule	151	(2)
7.5. The W^(k, p) Spaces	153	(1)
7.6. Density Theorems	154	(1)
7.7. Imbedding Theorems	155	(4)
7.8. Potential Estimates and Imbedding Theorems	159	(5)
7.9. The Morrey and John-Nirenberg Estimates	164	(3)
7.10. Compactness Results	167	(1)
7.11. Difference Quotients	168	(1)
7.12. Extension and Interpolation	169	(4)
Notes	173	(1)
Problems	173	(4)
Chapter 8. Generalized Solutions and Regularity	177	(42)
8.1. The Weak Maximum Principle	179	(2)
8.2. Solvability of the Dirichlet Problem	181	(2)
8.3. Differentiability of Weak Solutions	183	(3)
8.4. Global Regularity	186	(2)
8.5. Global Boundedness of Weak Solutions	188	(6)
8.6. Local Properties of Weak Solutions	194	(4)
8.7. The Strong Maximum Principle	198	(1)
8.8. The Harnack Inequality	199	(1)
8.9. Holder Continuity	200	(2)
8.10. Local Estimates at the Boundary	202	(7)
8.11. Holder Estimates for the First Derivatives	209	(3)
8.12. The Eigenvalue Problem	212	(2)
Notes	214	(2)
Problems	216	(3)
Chapter 9. Strong Solutions	219	(40)
9.1. Maximum Principles for Strong Solutions	220	(5)
9.2. L^p Estimates: Preliminary Analysis	225	(2)
9.3. The Marcinkiewicz Interpolation Theorem	227	(3)
9.4. The Calderon-Zygmund Inequality	230	(5)
9.5. L^p Estimates	235	(6)
9.6. The Dirichlet Problem	241	(3)
9.7. A Local Maximum Principle	244	(2)
9.8. Holder and Harnack Estimates	246	(4)
9.9. Local Estimates at the Boundary	250	(4)
Notes	254	(1)
Problems	255	(4)
Part II. Quasilinear Equations	259	(232)
Chapter 10. Maximum and Comparison Principles	259	(20)
10.1. The Comparison Principle	263	(1)
10.2. Maximum Principles	264	(3)
10.3. A Counterexample	267	(1)
10.4. Comparison Principles for Divergence Form Operators	268	(3)
10.5. Maximum Principles for Divergence Form Operators	271	(6)
Notes	277	(1)
Problems	277	(2)
Chapter 11. Topological Fixed Point Theorems and Their Application	279	(15)
11.1. The Schauder Fixed Point Theorem	279	(1)
11.2. The Leray-Schauder Theorem: a Special Case	280	(2)
11.3. An Application	282	(4)
11.4. The Leray-Schauder Fixed Point Theorem	286	(2)
11.5. Variational Problems	288	(5)
Notes	293	(1)
Chapter 12. Equations in Two Variables	294	(25)
12.1. Quasiconformal Mappings	294	(6)
12.2. Holder Gradient Estimates for Linear Equations	300	(4)
12.3. The Dirichlet Problem for Uniformly Elliptic Equations	304	(5)
12.4. Non-Uniformly Elliptic Equations	309	(6)
Notes	315	(2)
Problems	317	(2)
Chapter 13. Holder Estimates for the Gradient	319	(14)
13.1. Equations of Divergence Form	319	(4)
13.2. Equations in Two Variables	323	(1)
13.3. Equations of General Form; the Interior Estimate	324	(4)
13.4. Equations of General Form; the Boundary Estimate	328	(3)
13.5. Application to the Dirichlet Problem	331	(1)
Notes	332	(1)
Chapter 14. Boundary Gradient Estimates	333	(26)
14.1. General Domains	335	(2)
14.2. Convex Domains	337	(4)
14.3. Boundary Curvature Conditions	341	(6)
14.4. Non-Existence Results	347	(6)
14.5. Continuity Estimates	353	(1)
14.6. Appendix: Boundary Curvatures and the Distance Function	354	(3)
Notes	357	(1)
Problems	358	(1)
Chapter 15. Global and Interior Gradient Bounds	359	(29)
15.1. A Maximum Principle for the Gradient	359	(3)
15.2. The General Case	362	(7)
15.3. Interior Gradient Bounds	369	(4)
15.4. Equations in Divergence Form	373	(7)
15.5. Selected Existence Theorems	380	(4)
15.6. Existence Theorems for Continuous Boundary Values	384	(1)
Notes	385	(1)
Problems	386	(2)
Chapter 16. Equations of Mean Curvature Type	388	(53)
16.1. Hypersurfaces in R^(n+1)	388	(13)
16.2. Interior Gradient Bounds	401	(6)
16.3. Application to the Dirichlet Problem	407	(3)
16.4. Equations in Two Independent Variables	410	(3)
16.5. Quasiconformal Mappings	413	(10)
16.6. Graphs with Quasiconformal Gauss Map	423	(6)
16.7. Applications to Equations of Mean Curvature Type	429	(5)
16.8. Appendix: Elliptic Parametric Functionals	434	(3)
Notes	437	(1)
Problems	438	(3)
Chapter 17. Fully Nonlinear Equations	441	(50)
17.1. Maximum and Comparison Principles	443	(3)
17.2. The Method of Continuity	446	(4)
17.3. Equations in Two Variables	450	(3)
17.4. Holder Estimates for Second Derivatives	453	(10)
17.5. Dirichlet Problem for Uniformly Elliptic Equations	463	(4)
17.6. Second Derivative Estimates for Equations of Monge-Ampere Type	467	(4)
17.7. Dirichlet Problem for Equations of Monge-Ampere Type	471	(5)
17.8. Global Second Derivative Holder Estimates	476	(5)
17.9. Nonlinear Boundary Value Problems	481	(5)
Notes	486	(2)
Problems	488	(3)
Bibliography	491	(16)
Epilogue	507	(4)
Subject Index	511	(5)
Notation Index	516