Maximum principles and overdetermined problems for Hessian equations. (English) Zbl 1538.35100
Summary: 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).
MSC:
| 35B50 | Maximum principles in context of PDEs |
| 35B45 | A priori estimates in context of PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35J60 | Nonlinear elliptic equations |
| 35J96 | Monge-Ampère equations |
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