Abstract
In this note, as a particular case of a more general result, we obtain the following theorem.
Let be a nonempty bounded open set, and let be a continuous function which is in . Then, at least one of the following assertions holds:
(a) .
(b) There exists a nonempty open set , with , satisfying the following property: for every continuous function which is in , there exists such that, for each , the Jacobian determinant of the function vanishes at some point of .
As a consequence, if and is a nonnegative function, for each satisfying in the Monge–Ampère equation
one has
Citation
B. Ricceri. "The convex hull-like property and supported images of open sets." Ann. Funct. Anal. 7 (1) 150 - 157, February 2016. https://doi.org/10.1215/20088752-3428355
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