The Monge-Ampere Equation

by Gutierrez, Cristian E.

• ISBN13: 9780817641771
• ISBN10: 0817641777
• Format: Hardcover
• Copyright: 2001-04-01
• Publisher: Birkhauser

Summary

The classical Monge-Ampère equation has been the center of considerable interest in recent years because of its important role in several areas of applied mathematics. In reflecting these developments, this works stresses the geometric aspects of this beautiful theory, using some techniques from harmonic analysis---covering lemmas and set decompositions. Moreover, Monge-Ampère type equations have applications in the areas of differential geometry, the calculus of variations, and several optimization problems, such as the Monge-Kantorovitch mass transfer problem.The book is an essentially self-contained exposition of the theory of weak solutions, including the regularity results of L.A. Caffarelli. The presentation unfolds systematically from introductory chapters, and an effort is made to present complete proofs of all theorems. Included are examples, illustrations, bibliographical references at the end of each chapter, and a comprehensive index.Topics covered include: * Generalized Solutions * Non-divergence Equations * The Cross-Sections of Monge-Ampère * Convex Solutions of D^2u = 1 in R^n * Regularity Theory * W^2, p Estimates The Monge-Ampère Equation is a concise and useful book for graduate students and researchers in the field of nonlinear equations.

Table of Contents

Preface	ix
Notation	xi
Generalized Solutions to Monge-Ampere Equations	1	(30)
The normal mapping	1	(5)
Properties of the normal mapping	2	(4)
Generalized solutions	6	(2)
Viscosity solutions	8	(2)
Maximum principles	10	(7)
Aleksandrov's maximum principle	11	(1)
Aleksandrov-Bakelman-Pucci's maximum principle	12	(4)
Comparison principle	16	(1)
The Dirichlet problem	17	(2)
The nonhomogeneous Dirichlet problem	19	(5)
Return to viscosity solutions	24	(1)
Ellipsoids of minimum volume	25	(4)
Notes	29	(2)
Uniformly Elliptic Equations in Nondivergence Form	31	(14)
Critical density estimates	31	(6)
Estimate of the distribution function of solutions	37	(3)
Harnack's inequality	40	(3)
Notes	43	(2)
The Cross-sections of Monge-Ampere	45	(18)
Introduction	45	(2)
Preliminary results	47	(3)
Properties of the sections	50	(12)
The Monge-Ampere measures satisfying (3.1.1)	50	(5)
The engulfing property of the sections	55	(2)
The size of normalized sections	57	(5)
Notes	62	(1)
Convex Solutions of det D2u = 1 in Rn	63	(12)
Pogorelov's Lemma	63	(4)
Interior Holder estimates of D2u	67	(3)
Cα estimates of D2u	70	(4)
Notes	74	(1)
Regularity Theory for the Monge-Ampere Equation	75	(20)
External points	75	(2)
A result on extremal points of zeroes of solutions to Monge-Ampere	77	(2)
A strict convexity result	79	(5)
C1,α regularity	84	(8)
Examples	92	(1)
Notes	93	(2)
W2,p Estimates for the Monge-Ampere Equation	95	(28)
Approximation Theorem	95	(4)
Tangent paraboloids	99	(2)
Density estimates and power decay	101	(7)
Lp estimates of second derivatives	108	(4)
Proof of the Covering Theorem 6.3.3	112	(7)
Regularity of the convex envelope	119	(2)
Notes	121	(2)
Bibliography	123	(4)
Index	127