Non-transversal intersection of free and fixed boundary for fully nonlinear elliptic operators in two dimensions

TL;DR

This paper proves that in two-dimensional fully nonlinear elliptic obstacle problems, the free boundary tangentially touches the fixed boundary, extending classical results beyond linear operators and providing new regularity and blow-up analysis insights.

Contribution

It introduces a novel approach to establish tangential contact in fully nonlinear settings, overcoming limitations of linear-based methods.

Findings

Free boundary tangentially touches fixed boundary in 2D for nonlinear operators

Established $C^{1,1}$ regularity up to fixed boundary

Derived new $n$-dimensional estimates and blow-up behavior descriptions

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 boundary 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.