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The Monge-Ampere Equation - Cristian E. Gutierrez

The Monge-Ampere Equation

Buch | Hardcover
143 Seiten
2001 | 2001 ed.
Birkhauser Boston Inc (Verlag)
978-0-8176-4177-1 (ISBN)
CHF 194,70 inkl. MwSt
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The Monge-Ampere equation has attracted considerable interest in years because of its important role in several areas of applied mathematics. This book stresses the geometric aspects of this beautiful theory, using techniques from harmonic analysis - covering lemmas and set decompositions.
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.

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-Ampere 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 Holder estimates of D2u.- 4.3 C?estimates of D2u.- 4.4 Notes.- 5 Regularity Theory for the Monge-Ampere Equation.- 5.1 Extremal points.- 5.2 A result on extremal points of zeroes of solutions to Monge-Ampere.- 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.

Reihe/Serie Progress in Nonlinear Differential Equations and Their Applications ; 44
Zusatzinfo biography
Verlagsort Secaucus
Sprache englisch
Maße 156 x 234 mm
Gewicht 387 g
Themenwelt Mathematik / Informatik Mathematik Analysis
Mathematik / Informatik Mathematik Angewandte Mathematik
Mathematik / Informatik Mathematik Geometrie / Topologie
ISBN-10 0-8176-4177-7 / 0817641777
ISBN-13 978-0-8176-4177-1 / 9780817641771
Zustand Neuware
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