Nontransversal intersection of free and fixed boundaries for fully nonlinear elliptic operators in two dimensions

Abstract

In the study of classical obstacle problems, it is well known that in many configurations, the free boundary intersects the fixed boundary tangentially. The arguments involved in producing results of this type rely on the linear structure of the operator. In this paper, we employ a different approach and prove tangential touch of free and fixed boundaries in two dimensions for fully nonlinear elliptic operators. Along the way, several $n$-dimensional results of independent interest are obtained, such as BMO-estimates, $C^{1,1}$-regularity up to the fixed boundary, and a description of the behavior of blow-up solutions.