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

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