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.
Ibrokhimbek AKRAMOV
,
Marcel OLIVER
. ON THE EXISTENCE OF SOLUTIONS TO A BI-PLANAR MONGE-AMPÈRE EQUATION[J]. Acta mathematica scientia, Series B, 2020
, 40(2)
: 379
-388
.
DOI: 10.1007/s10473-020-0206-6
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