The Monge-Ampere Equation
Birkhauser Boston Inc (Verlag)
978-0-8176-4177-1 (ISBN)
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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 |
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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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