数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (2): 379-388.doi: 10.1007/s10473-020-0206-6

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ON THE EXISTENCE OF SOLUTIONS TO A BI-PLANAR MONGE-AMPÈRE EQUATION

Ibrokhimbek AKRAMOV1, Marcel OLIVER2   

  1. 1. Institut für Angewandte Analysis, Universität Ulm, 89069 Ulm, Germany;
    2. School of Engineering and Science, Jacobs University, 28759 Bremen, Germany
  • 收稿日期:2018-05-22 修回日期:2019-03-21 出版日期:2020-04-25 发布日期:2020-05-26
  • 作者简介:Ibrokhimbek AKRAMOV,E-mail:ibrokhimbek.akramov@uni-ulm.de;Marcel OLIVER,E-mail:m.oliver@jacobs-university.de
  • 基金资助:
    This article contributes to the project "Systematic multi-scale modeling and analysis for geophysical flow" of the Collaborative Research Center TRR 181 "Energy Transfers in Atmosphere and Ocean" funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under project number 274762653.

ON THE EXISTENCE OF SOLUTIONS TO A BI-PLANAR MONGE-AMPÈRE EQUATION

Ibrokhimbek AKRAMOV1, Marcel OLIVER2   

  1. 1. Institut für Angewandte Analysis, Universität Ulm, 89069 Ulm, Germany;
    2. School of Engineering and Science, Jacobs University, 28759 Bremen, Germany
  • Received:2018-05-22 Revised:2019-03-21 Online:2020-04-25 Published:2020-05-26
  • Supported by:
    This article contributes to the project "Systematic multi-scale modeling and analysis for geophysical flow" of the Collaborative Research Center TRR 181 "Energy Transfers in Atmosphere and Ocean" funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under project number 274762653.

摘要: In this article, we consider a fully nonlinear partial differential equation which can be expressed as a sum of two Monge-Ampère operators acting in different two-dimensional coordinate sections. This equation is elliptic, for example, in the class of convex functions. We show that the notion of Monge-Ampère measures and Aleksandrov generalized solutions extends to this equation, subject to a weaker notion of convexity which we call bi-planar convexity. While the equation is also elliptic in the class of bi-planar convex functions, the contrary is not necessarily true. This is a substantial difference compared to the classical Monge-Ampère equation where ellipticity and convexity coincide. We provide explicit counter-examples: classical solutions to the bi-planar equation that satisfy the ellipticity condition but are not generalized solutions in the sense introduced. We conclude that the concept of generalized solutions based on convexity arguments is not a natural setting for the bi-planar equation.

关键词: Fully nonlinear elliptic equations, generalized solution, bi-planar convexity

Abstract: In this article, we consider a fully nonlinear partial differential equation which can be expressed as a sum of two Monge-Ampère operators acting in different two-dimensional coordinate sections. This equation is elliptic, for example, in the class of convex functions. We show that the notion of Monge-Ampère measures and Aleksandrov generalized solutions extends to this equation, subject to a weaker notion of convexity which we call bi-planar convexity. While the equation is also elliptic in the class of bi-planar convex functions, the contrary is not necessarily true. This is a substantial difference compared to the classical Monge-Ampère equation where ellipticity and convexity coincide. We provide explicit counter-examples: classical solutions to the bi-planar equation that satisfy the ellipticity condition but are not generalized solutions in the sense introduced. We conclude that the concept of generalized solutions based on convexity arguments is not a natural setting for the bi-planar equation.

Key words: Fully nonlinear elliptic equations, generalized solution, bi-planar convexity

中图分类号: 

  • 35J60