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Problems for generalized Monge–Ampère equations

Published online by Cambridge University Press:  11 September 2023

Cristian Enache*
Affiliation:
Department of Mathematics and Statistics, American University of Sharjah, Sharjah, U.A.E.
Giovanni Porru
Affiliation:
Dipartimento di Matematica e Informatica, University of Cagliari, Cagliari, Italy e-mail: porru@unica.it
*

Abstract

This paper deals with some Monge–Ampère type equations involving the gradient that are elliptic in the framework of convex functions. First, we show that such equations may be obtained by minimizing a suitable functional. Moreover, we investigate a P-function associated with the solution to a boundary value problem of our generalized Monge–Ampère equation in a bounded convex domain. It will be shown that this P-function attains its maximum value on the boundary of the underlying domain. Furthermore, we show that such a P-function is actually identically constant when the underlying domain is a ball. Therefore, our result provides a best possible maximum principles in the sense of L. E. Payne. Finally, in case of dimension 2, we prove that this P-function also attains its minimum value on the boundary of the underlying domain. As an application, we will show that the solvability of a Serrin’s type overdetermined problem for our generalized Monge–Ampère type equation forces the underlying domain to be a ball.

Type
Article
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

Dedicated to Prof. Gérard A. Philippin on the occasion of his 80th birthday.

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