The Monge--Ampère Equation

ISBN-13: 9781461266563 / Angielski / Miękka / 2012 / 132 str.

Cristian E. Gutierrez; Cristian E. Gutierrez

Opis

The Monge-Ampere equation has attracted considerable interest in recent years because of its important role in several areas of applied mathematics. Monge-Ampere 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. This book stresses the geometric aspects of this beautiful theory, using techniques from harmonic analysis - covering lemmas and set decompositions.

Spis treści

1 Generalized Solutions to Monge-Ampere Equations.- 1.1 The normal mapping.- 1.1.1 Properties of the normal mapping.- 1.2 Generalized solutions.- 1.3 Viscosity solutions.- 1.4 Maximum principles.- 1.4.1 Aleksandrov’s maximum principle.- 1.4.2 Aleksandrov-Bakelman-Pucci’s maximum principle.- 1.4.3 Comparison principle.- 1.5 The Dirichlet problem.- 1.6 The nonhomogeneous Dirichlet problem.- 1.7 Return to viscosity solutions.- 1.8 Ellipsoids of minimum volume.- 1.9 Notes.- 2 Uniformly Elliptic Equations in Nondivergence Form.- 2.1 Critical density estimates.- 2.2 Estimate of the distribution function of solutions.- 2.3 Harnack’s inequality.- 2.4 Notes.- 3 The Cross-sections of Monge-Ampere.- 3.1 Introduction.- 3.2 Preliminary results.- 3.3 Properties of the sections.- 3.3.1 The Monge-Ampère measures satisfying (3.1.1).- 3.3.2 The engulfing property of the sections.- 3.3.3 The size of normalized sections.- 3.4 Notes.- 4 Convex Solutions of det D2u = 1 in ?n.- 4.1 Pogorelov’s Lemma.- 4.2 Interior Hölder estimates of D2u.- 4.3 C?estimates of D2u.- 4.4 Notes.- 5 Regularity Theory for the Monge-Ampère Equation.- 5.1 Extremal points.- 5.2 A result on extremal points of zeroes of solutions to Monge-Ampère.- 5.3 A strict convexity result.- 5.4 C1,?regularity.- 5.5 Examples.- 5.6 Notes.- 6 W2pEstimates for the Monge-Ampere Equation.- 6.1 Approximation Theorem.- 6.2 Tangent paraboloids.- 6.3 Density estimates and power decay.- 6.4 LP estimates of second derivatives.- 6.5 Proof of the Covering Theorem 6.3.3.- 6.6 Regularity of the convex envelope.- 6.7 Notes.

O autorze

Reflects recent developments related to the Monge-Ampere equation stressing the geometric aspects of the theory, using some techniques from harmonic analysis---covering lemmas and set decompositions. Essentially self-contained, 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. A concise and useful book for graduate students and researchers in the field of nonlinear equations.