-
I. Akramov and M. Oliver,
On the existence of solutions to a bi-planar Monge-Ampère equation,
Acta Math. Sci. 40 (2020), 379-388.
Abstract:
In this paper, 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
Alexandrov 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.
Download the paper in
PDF
format.