| Preface Notation 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 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 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 4 Convex Solutions of det D[superscript 2]u = 1 in R[superscript n] 4.1 Pogorelov's Lemma 4.2 Interior Holder estimates of D[superscript 2]u u 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 C[superscript 1,[alpha]] regularity 5.5 Examples 6 W[superscript 2,p] Estimates for the Monge-Ampere Equation 6.1 Approximation Theorem 6.2 Tangent paraboloids 6.3 Density estimates and power decay 6.4 L[superscript p] estimates of second derivatives 6.5 Proof of the Covering Theorem 6.3.3 6.6 Regularity of the convex envelope Bibliography Index |
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