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Problems for generalized Monge–Ampère equations

Published online by Cambridge University Press:  11 September 2023

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


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.

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

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


Barbu, L. and Enache, C., A maximum principle for some fully nonlinear elliptic equations with applications to Weingarten hypersurfaces . Complex Var. Elliptic Equ. 58(2013), no. 12, 17251736.CrossRefGoogle Scholar
Brandolini, B., Nitsch, C., Salani, P., and Trombetti, C., Serrin-type overdetermined problems: an alternative proof . Arch. Ration. Mech. Anal. 190(2008), no. 2, 267280.CrossRefGoogle Scholar
Caffarelli, L., Nirenberg, L., and Spruck, J., The Dirichlet problem for nonlinear second order elliptic equations III: functions of eigenvalues of the Hessian . Acta Math. 155(1985), 261301.CrossRefGoogle Scholar
Chen, C., Ma, X., and Shi, S., Curvature estimates for the level sets of solutions to the Monge–Ampère equation ${\mathrm{D}}^2\mathrm{u}=1$ . Chin. Ann. Math. Ser. B 35(2014), 895906.CrossRefGoogle Scholar
Chou, K.-S. and Wang, X.-J., A variational theory of the Hessian equation . Commun. Pure Appl. Math. 54(2001), no. 9, 10291064.CrossRefGoogle Scholar
Enache, C., Maximum and minimum principles for a class of Monge–Ampère equations in the plane, with applications to surfaces of constant gauss curvature . Commun. Pure Appl. Anal. 13(2014), no. 3, 13471359.CrossRefGoogle Scholar
Enache, C., Necessary conditions of solvability and isoperimetric estimates for some Monge–Ampère problems in the plane . Proc. Amer. Math. Soc. 143(2015), no. 1, 309315.CrossRefGoogle Scholar
Enache, C., Marras, M., and Porru, G., Maximum principles and overdetermined problems for Hessian equations . Adv. Pure Appl. Math. 12(2021), no. spécial, 123138.CrossRefGoogle Scholar
Horn, R. A. and Johnson, C. R., Matrix analysis. 2nd ed., Cambridge University Press, Cambridge, 2013.Google Scholar
Mohammed, A. and Porru, G., A sharp global estimate and an overdetermined problem for Monge–Ampère type equations . Adv. Nonlinear Stud. 22(2022), 114.CrossRefGoogle Scholar
Payne, L. E., Some applications of “best possible” maximum principles in elliptic boundary value problems, Research notes in Math. 101, Pitman, Boston-London (1984), 286313.Google Scholar
Philippin, G. A., Applications of the maximum principle to a variety of problems involving elliptic differential equations. Maximum principles and eigenvalue problems in partial differential equations (Knoxville, TN, 1987), Pitman Res. Notes Math. Ser., 175, Longman Sci. Tech., Harlow, (1988), 3448.Google Scholar
Porru, G., Safoui, A., and Vernier-Piro, S., Best possible maximum principles for fully nonlinear elliptic partial differential equations . Z. Anal. Anwend. 25(2006), 421434.CrossRefGoogle Scholar
Reilly, R. C., On the hessian of a function and the curvatures of its graph . Michigan Math. J. 20(1973), 373383.Google Scholar
Reilly, R. C., Variational properties of functions of the mean curvatures for hypersurfaces in space forms . J. Differential Geometry 8(1973), 465477.CrossRefGoogle Scholar
Serrin, J., A symmetry problem in potential theory . Arch. Ration. Mech. Anal. 43(1971), 304318.CrossRefGoogle Scholar
Tso, K., On a real Monge–Ampère functional . Invent. Math. 101(1990), 425448.CrossRefGoogle Scholar
Weinberger, H. F., Remark on the preceding paper of Serrin . Arch. Ration. Mech. Anal. 43(1971), 319328.CrossRefGoogle Scholar