A maximum principle for some fully nonlinear elliptic equations with applications to Weingarten hypersurfaces

Abstract

In this article, we investigate a general class of fully nonlinear elliptic equations, including Weingarten equations. Our first aim is to construct a general elliptic inequality for an appropriate functional combination of u(x) and |∇u(x)|, i.e. a kind of P-function P(x), in the sense of L.E. Payne (see the book of Sperb [Sperb, Maximum Principles and Their Applications, Academic Press, New York, 1981]), where u(x) is a given solution of our class of fully nonlinear equations. From this inequality, making use of Hopf's first maximum principle, we derive a maximum principle for P(x), which extend some similar results obtained by Philippin and Safoui [Philippin and Safoui, Some maximum principles and symmetry results for a class of boundary value problems involving the Monge-Ampère equation, Math. Models Methods Appl. Sci. 11 (2001), pp. 1073–1080; Philippin and Safoui, Some applications of the maximum principle to a variety of fully nonlinear elliptic PDE's, Z. Angew. Math. Phys. 54 (2003), pp. 739–755], Porru et al. [Porru, Safoui and Vernier-Piro, Best possible maximum principles for fully nonlinear elliptic partial differential equations, Zeit. Anal. Anwend. 25 (2006), pp. 421–434] and Enache [Enache, Maximum principles and symmetry results for a class of fully nonlinear elliptic PDEs, Nonlinear Differ. Eqns Appl. 17 (2010), pp. 591–600]. This maximum principle is then used to obtain various a priori estimates with applications to some class of Weingarten hypersurfaces.

Keywords:

• fully nonlinear elliptic equations
• maximum principles
• a priori bounds
• Weingarten hypersurfaces

AMS Subject Classifications::

• 35B50
• 35J60
• 53A10

Acknowledgements

The authors thank the anonymous referees for their valuable suggestions to improve clarity in several points. The second author was supported by the strategic grant POSDRU/88/1.5/S/49516 Project ID 49516 (2009), co-financed by the European Social Fund Investing in People, within the Sectorial Operational Programme Human Resources Development 2007–2013.