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Maximum principles and overdetermined problems for Hessian equations
Enache, Cristian Marras, Monica Porru, Giovanni
Enache, Cristian
Marras, Monica
Porru, Giovanni
Date
2021
Advisor
Type
Article
Peer-Reviewed
Published version
Peer-Reviewed
Published version
Degree
Description
Abstract
In this article we investigate some Hessian type equations. Our main aim is to derive new maximum principles for some suitable P-functions, in the sense of L.E. Payne, that is for some appropriate functional combinations of u(x) and its derivatives, where u(x) is a solution of the given Hessian type equations. To find the most suitable P-functions, we first investigate the special case of a ball, where the solution of our Hessian equations is radial, since this case gives good hints on the best functional to be considered later, for general domains. Next, we construct some elliptic inequalities for the well-chosen P-functions and make use of the classical maximum principles to get our new maximum principles. Finally, we consider some overdetermined problems and show that they have solutions when the underlying domain has a certain shape (spherical or ellipsoidal).