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＜電子ブック＞
The Monge—Ampère Equation / by Cristian E. Gutierrez
(Progress in Nonlinear Differential Equations and Their Applications. ISSN:23740280 ; 44)

版	1st ed. 2001.
出版者	(Boston, MA : Birkhäuser Boston : Imprint: Birkhäuser)
出版年	2001
本文言語	英語
大きさ	XI, 132 p : online resource
著者標目	*Gutierrez, Cristian E author SpringerLink (Online service)
件　名	LCSH:Differential equations LCSH:Mathematics LCSH:Geometry, Differential FREE:Differential Equations FREE:Applications of Mathematics FREE:Differential Geometry
一般注記	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 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 HTTP:URL=https://doi.org/10.1007/978-1-4612-0195-3

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