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

  • Guido De Philippis

    New York University, New York, USA
  • Luca Spolaor

    University of California, San Diego, La Jolla, USA
  • Bozhidar Velichkov

    Università di Pisa, Pisa, Italy
(Quasi-)conformal methods in two-dimensional free boundary problems cover

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Abstract

In this paper, 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 apply this result to the solutions of one-phase Bernoulli free boundary 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 this article

Guido De Philippis, Luca Spolaor, Bozhidar Velichkov, (Quasi-)conformal methods in two-dimensional free boundary problems. J. Eur. Math. Soc. (2024), published online first

DOI 10.4171/JEMS/1435