On the Existence of Solutions to a Bi-Planar Monge-Ampère Equation
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 Mon...
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Veröffentlicht in: | Acta mathematica scientia 2020-03, Vol.40 (2), p.379-388 |
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Sprache: | eng |
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Zusammenfassung: | 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. |
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ISSN: | 0252-9602 1572-9087 |
DOI: | 10.1007/s10473-020-0206-6 |