A Bernoulli free-boundary problem is one of finding domains in the plane on which a harmonic function simultaneously satisfies linear homogeneous Dirichlet and inhomogeneous Neumann boundary conditions. For a general class of Bernoulli problems, we prove that any free boundary, possibly with many singularities, is necessarily the graph of a function. Also investigated are convexity and monotonicity properties of free boundaries. In addition, we obtain some optimal estimates on the gradient of the harmonic function in question.
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Eugen Varvaruca, Some geometric and analytic properties of solutions of Bernoulli free-boundary problems. Interfaces Free Bound. 9 (2007), no. 3, pp. 367–381DOI 10.4171/IFB/169