Author | Gutiรฉrrez, Cristian E. author |
---|---|
Title | The MongeโAmpรจre Equation [electronic resource] / by Cristian E. Gutiรฉrrez |
Imprint | Boston, MA : Birkhรคuser Boston : Imprint: Birkhรคuser, 2001 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-0195-3 |
Descript | XI, 132 p. online resource |
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 Harnackโs 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 Pogorelovโs Lemma -- 4.2 Interior Hรถlder estimates of D2u -- 4.3 C?estimates of D2u -- 4.4 Notes -- 5 Regularity Theory for the Monge-Ampรจre Equation -- 5.1 Extremal points -- 5.2 A result on extremal points of zeroes of solutions to Monge-Ampรจre -- 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