The geometry of the free boundary

TL;DR

This paper advances understanding of free boundary regularity in nonlinear elliptic problems, demonstrating non-transversal intersections and $C^1$ regularity near fixed boundaries, with novel geometric insights into boundary convergence.

Contribution

It establishes non-transversal intersection results and $C^1$ regularity of free boundaries in nonlinear elliptic problems, extending known results from the Laplacian case to higher dimensions.

Findings

Non-transversal intersection of free and fixed boundaries for nonlinear operators.

$C^1$ regularity of free boundary near fixed boundary in 2D.

New geometric approach to boundary point convergence and spacing.

Abstract

The non-transversal intersection of the free boundary with the fixed boundary is obtained for nonlinear uniformly elliptic operators when $\Omega = \{\nabla u \neq 0\} \cap \{x_n>0\}$ thereby solving a problem in elliptic theory that in the case of the Laplacian is completely understood but has remained arcane in the nonlinear setting in higher dimension. Also, a solution is given to a problem discussed in "Regularity of free boundaries in obstacle-type problems" \cite{MR2962060}. The free boundary is $C^1$ in a neighborhood of the fixed if the solution is physical and if $n=2$ in the absolute general context. The regularity is even new for the Laplacian. The innovation is via geometric configurations on how free boundary points converge to the fixed boundary and investigating the spacing between free boundary points.