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Necessary conditions of solvability and isoperimetric estimates for some Monge-Ampère problems in the plane

Author: Cristian Enache
Journal: Proc. Amer. Math. Soc. 143 (2015), 309-315
MSC (2010): Primary 35J60, 35J96, 35J25, 35B50; Secondary 53C45
Published electronically: September 24, 2014
MathSciNet review: 3272756
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Abstract: This article is mainly devoted to the solvability of the
Monge-Ampère equation $ \det \left ( D^{2}u\right ) =1,$ in a $ C^{2}$ bounded strictly convex domain $ \Omega \subset \mathbb{R}^{2}$, subject to a contact angle boundary condition. A necessary condition for the solvability of this problem, involving the maximal value of the curvature $ k\left ( s\right ) $ of $ \partial \Omega $ and the contact angle, was derived by X.-N. Ma in 1999, making use of a maximum principle for an appropriate P-function. Our main goal here is to prove a complementary result. More precisely, we will derive a new necessary condition of solvability, involving the minimal value of the curvature $ k\left ( s\right ) $ of $ \partial \Omega $ and the contact angle. The main ingredients of our proof are the derivation of a minimum principle for the P-function employed by X.-N. Ma in his proof, respectively, the use of some computations in normal coordinates with respect to the boundary $ \partial \Omega $. Finally, a similar minimum principle will be employed to derive some isoperimetric estimates for the classical convex solution of the Monge-Ampère equation, subject to the homogeneous Dirichlet boundary condition.

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Additional Information

Cristian Enache
Affiliation: Department of Mathematics and Informatics, Ovidius University of Constanta, Constanta, 900527, Romania

Received by editor(s): January 8, 2013
Received by editor(s) in revised form: April 7, 2013
Published electronically: September 24, 2014
Additional Notes: The author was supported by the strategic grant POSDRU/88/1.5/S/49516 Project ID 49516 (2009), co-financed by the European Social Fund Investing in People, within the Sectorial Operational Programme Human Resources Development 2007–2013.
Communicated by: Joachim Krieger
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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