The Monge—Ampère Equation

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.