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Bernoulli free-boundary problems. (English) Zbl 1167.35001

Mem. Am. Math. Soc. 914, 70 p. (2008).
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. The boundary is called “free” because it is not prescribed a priori, and the problem of determining free boundaries from the given data is nonlinear. The name Bernoulli was originally associated which such problems in hydrodynamics. Questions of existence, multiplicity or uniqueness, and regularity of free boundaries for prescribed data are very important and their solutions lead to nonlinear problems. In the booklet an equivalence is shown between Bernoulli free-boundary problems and a class of equations for real-valued functions of one real variable. Since no restriction is imposed by the authors on the amplitudes or shapes of free boundaries, this equivalence is global, and valid even for very weak solutions.
Furthermore it is shown in the paper that the above equivalence can be expressed as nonlinear Riemann-Hilbert problems and the theory of Hardy spaces in the unit disc plays a central role. Also, they have gradient structure and their solutions are critical points of a natural Lagrangian. Therefore, the Calculus of Variations can successfully be applied as a tool for the study. Some rather natural conjectures about the regularity of free boundaries remain unresolved.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35R35 Free boundary problems for PDEs
76B07 Free-surface potential flows for incompressible inviscid fluids
35Q15 Riemann-Hilbert problems in context of PDEs
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
35B65 Smoothness and regularity of solutions to PDEs
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