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Full Description
In recent years, the study of the Monge-Ampere equation has received consider able attention and there have been many important advances. As a consequence there is nowadays much interest in this equation and its applications. This volume tries to reflect these advances in an essentially self-contained systematic exposi tion of the theory of weak: solutions, including recent regularity results by L. A. Caffarelli. The theory has a geometric flavor and uses some techniques from har monic analysis such us covering lemmas and set decompositions. An overview of the contents of the book is as follows. We shall be concerned with the Monge-Ampere equation, which for a smooth function u, is given by (0.0.1) There is a notion of generalized or weak solution to (0.0.1): for u convex in a domain n, one can define a measure Mu in n such that if u is smooth, then Mu 2 has density det D u. Therefore u is a generalized solution of (0.0.1) if M u = f.
Contents
1 Generalized Solutions to Monge-Ampere Equations.- 1.1 The normal mapping.- 1.2 Generalized solutions.- 1.3 Viscosity solutions.- 1.4 Maximum principles.- 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.- 3.4 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.
