(Quasi-)conformal methods in two-dimensional free boundary problems
Analysis of PDEs
2021-10-28 v1 Complex Variables
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.
Cite
@article{arxiv.2110.14075,
title = {(Quasi-)conformal methods in two-dimensional free boundary problems},
author = {Guido De Philippis and Luca Spolaor and Bozhidar Velichkov},
journal= {arXiv preprint arXiv:2110.14075},
year = {2021}
}