Extremality conditions for the quasi-concavity function and applications

A maximum principle for the lower envelope of two strictly subharmonic functions is proved, and subsequently used to investigate the first- and second-order extremality conditions for the quasi-concavity function. An application is done to the Dirichlet problem associated to elliptic equations involving the Laplacian as well as the minimal surface operator, when the domain of the problem is a convex ring and two constant boundary values are prescribed. The right-hand side may depend on the solution and on any of its first derivatives, and must depend on the space variable. The solution is proved to have convex level sets and a non-vanishing gradient. Assumptions are translation-invariant. Poisson’s equation is considered explicitly.