Behavior of the Free Boundary Near Contact Points with the Fixed Boundary for Nonlinear Elliptic Equations

Abstract

Abstract. The aim of this paper is to study a free boundary problem for a uniformly elliptic fully non-linear operator. Under certain assumptions we show that free and fixed boundaries meet tangentially at contact points.

Boundary Regularity and Nontransversal Intersection for the Fully Nonlinear Obstacle Problem

“…It has been conjectured that the free boundary intersects the fixed boundary nontransversally and in two dimensions this was proved in [IM16a] (partial results have also been obtained in [MM04]). The case of the Laplacian was treated in [SU03,AU95].…”

Free boundary regularity near the fixed boundary for the fully nonlinear obstacle problem

“…In particular, it must be a half-space solution by Proposition 2.3 and the non-transversal intersection follows as before. The assumption on the negativity set appeared in [MM04] where the authors considered the non-transversal intersection subject to additional assumptions on the operator and solution.…”

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

“…Under further thickness assumptions, these results imply C 1 regularity of the free boundary. However, a description of the dynamics on how the free boundaries intersect the fixed boundary has remained an open problem for at least a decade in the fully nonlinear setting (although partial results have been obtained in [MM04] under strong density and growth assumptions involving the solutions and a homogeneity assumption on the operator). On the other hand, extensive work has been carried out to investigate this question for the classical problem ∆u = χ u>0 in B 1 ∩ {x n > 0}, u = 0 on {x n = 0}, and its variations [AU95, SU03, Mat05, AMM06, And07].…”