# Nonlinear Analysis on Manifolds: Monge-Ampere Equations

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This material is useful in other fields of mathematics, such as partial differential equations, to name one. We feel that workers in PDE would be more comfortable with the covariant derivative if they had studied it in a course such as the present one. If you're assuming the Riemannian manifold has a fixed metric, then the most introductory source I've found is Folland, Introduction to Partial Differential Equations , which discusses aspects of PDEs on hypersurfaces and the Laplace-Beltrami operator, for example. Sign up to join this community. The best answers are voted up and rise to the top. Asked 7 years, 2 months ago. Active 6 years, 4 months ago. Viewed 2k times. Also, any topics to particularly study or avoid would be useful. Horton Jul 15 '12 at An excerpt from its preface: The aim of this book is to facilitate the teaching of differential geometry. A good book by a late master,although terse and challenging. If you like that kind of presentation a la Warner or Walshap, then this is one of the best.

Sign up or log in Sign up using Google. In Section 2 we will collect some known results to be used in the subsequent sections. Section 3 is devoted to the proofs of the Theorem 1. In Section 4 we prove that Theorem 1. In Appendix we will introduce the theory of regular variation for the proof of Corollary 1. Lemma 2. This observation will be used later in the paper. Proposition 2. The existence result below is a variant of Lemma 2. It follows that there exist positive constants b 1 and b 2 such that. It was proved in [ 34 ]. Assume to the contrary that 1. We aim to derive a contradiction.

Then define. Let w 1 x be a strictly convex solution of 1. Since w is strictly convex, all the eigenvalue of B w is positive. We thus obtain.

This contradiction completes our proof. It follows from Theorem 1. Let A be large enough such that. By Lemma 2.

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It follows from Proposition AP. For the proof of Theorem 1.

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In the radially symmetric setting, the smoothness requirements for K and f can be greatly relaxed. But for convenience, we still use K , K1 and f1. In the case K1 can be state as:.

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Suppose that K satisfies K and K1. Similar to the proof of Theorem 5. So we omit it here. Then by 1.

## Monge-Ampère equation - Encyclopedia of Mathematics

By the definition of z we have z x i x j is negative definite. Fix m.

By Lemma 4. Then by 2. It follows from Lemma 2. Thus u is a strictly convex solution of 1. We present some basic facts of Karamata regular variation theory refer to [ 60 ], [ 61 ] here. Uniform convergence theorem. Representation theorem. A function L is slowly varying at infinity if and only if it may be written in the form. The result of Proposition AP. The way to remember roposition AP. Let f satisfy f1 and 1. So we omit it. Let f satisfy f1 , 1. We have. It can be proved by the definition of regularly varying and rapidly varying at infinity.

The authors are grateful to anonymous referees for their constructive comments and suggestions, which has greatly improved this paper. Trudinger and X. Caffarelli, J. KOHN, L. Nirenberg, J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations II.

Optimal transportation, Monge-Ampère equations and Riemannian Ricci bounds - 1

Pure Appl. Equations, , , Gilbarg and N.