Abstract
This paper investigates a P-function associated with solutions to boundary value problems of some generalized Monge-Ampère equations in bounded convex domains. It will be shown that P attains its maximum value either on the boundary or at a critical point of any convex solution. Furthermore, it turns out that such P-function is actually a constant when the underlying domain is a ball. Therefore, our results provide a best possible maximum principle in the sense of L. Payne. As an application, we will use these results to study an overdetermined boundary value problem. More specifically, we will show solvability of this overdetermined boundary value problem forces their P-function to be a constant.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Caffarelli, L., Nirenberg, L., Spruck, J.: The Dirichlet problem for nonlinear second order elliptic equations I: Monge-Ampère equation. Commun. Pure Appl. Math. 37(3), 369–402 (1984)
Chen, C., Ma, X., Shi, S.: Curvature estimates for the level sets of solutions to the Monge-Ampère equation det\((D^2u) = 1\). Chin. Ann. Math. Ser. B 35(6), 895–906 (2014)
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, 1347–1359 (2014)
Figalli, A.: The Monge-Ampère Equation and Its Applications. Zurich Lectures in Advanced Mathematics
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1983)
Mohammed, A., Porru, G., Safoui, A.: Isoperimetric inequalities for Hessian equations. Adv. Nonlinear Anal. 1, 181–203 (2012)
Payne, L.E.: Isoperimetric inequalities and their applications. SIAM Rev. 9, 453–488 (1967)
Payne, L.E.: Some applications of “best possible" maximum principles in elliptic boundary value problems. Res. Notes Math. 101, 286–313 (1984)
Porru, G., Safoui, A., Vernier-Piro, S.: Best possible maximum principles for fully nonlinear elliptic partial differential equations. Z. Anal. Anwend. 25, 421–434 (2006)
Reilly, R.C.: On the Hessian of a function and the curvatures of its graph. Mich. Math. J. 20, 373–383 (1973)
Reilly, R.C.: Variational properties of functions of the mean curvatures for hypersurfaces in space forms. J. Differ. Geom. 8, 465–477 (1973)
Serrin, J.: A symmetry problem in potential theory. Arch. Rat. Mech. Anal. 43, 304–318 (1971)
Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 51, 126–150 (1984)
Trudinger, N.S.: On new isoperimetric inequalities and symmetrization. J. Reine Angew. Math. 488, 203–220 (1997)
Weinberger, H.: Remark on the preceding paper of Serrin. Arch. Rat. Mech. Anal. 43, 319–320 (1971)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Mohammed, A., Porru, G. A Best Possible Maximum Principle and an Overdetermined Problem for a Generalized Monge-Ampère Equation. J Geom Anal 34, 49 (2024). https://doi.org/10.1007/s12220-023-01500-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12220-023-01500-w
Keywords
- Monge-Ampère type equations
- P-function
- Best possible maximum principle
- Overdetermined boundary-value problem