(Quasi-)conformal methods in two-dimensional free boundary problems

TL;DR

This paper explores the use of quasi-conformal maps to analyze the local behavior and regularity of solutions in two-dimensional free boundary problems, introducing new transforms and characterizations of singularities.

Contribution

It introduces a conformal hodograph transform and relates quasi-conformal map theory to free boundary regularity, providing new insights into solution structure and singularities.

Findings

Contact set is a finite union of intervals

Precise expansion at branch points obtained

Examples of cusp-like free boundary singularities constructed

Abstract

In this paper we study the local behavior of solutions to some free boundary problems. We relate the theory of quasi-conformal maps to the regularity of the solutions to nonlinear thin-obstacle problems; we prove that the contact set is locally a finite union of intervals and we apply this result to the solutions of one-phase Bernoulli problems with geometric constraint. We also introduce a new conformal hodograph transform, which allows to obtain the precise expansion at branch points of both the solutions to the one-phase problem with geometric constraint and a class of symmetric solutions to the two-phase problem, as well as to construct examples of free boundaries with cusp-like singularities.