Maximum principles and overdetermined problems for Hessian equations

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).

Problems for generalized Monge–Ampère equations

“…As a corollary, we solve a Serrin’s type overdetermined boundary value problem (see [2, 16, 18]) for the corresponding g -Monge–Ampère equation. Similar problems are discussed in [1, 8, 10, 13] and the references therein.…”

“…By Theorem 2.3 of [8], P(x) attains its maximum value on ∂Ω; furthermore, in case Ω is a ball, P(x) is a constant. We are going to extend this result to our g-Monge-Ampère equation.…”

“…For a discussion on the best possible maximum principles related to second-order linear (or quasi-linear) elliptic equations, we refer to [12]. Concerning Monge–Ampère equations, we recall a special case of Theorem 2.3 of [8]. Let u be a strictly convex smooth solution to the problem and let By Theorem 2.3 of [8], attains its maximum value on ; furthermore, in case is a ball, is a constant.…”

“…Concerning Monge–Ampère equations, we recall a special case of Theorem 2.3 of [8]. Let u be a strictly convex smooth solution to the problem and let By Theorem 2.3 of [8], attains its maximum value on ; furthermore, in case is a ball, is a constant.…”