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On a Bernoulli problem with geometric constraints. (English) Zbl 1337.49004

Summary: A Bernoulli free boundary problem with geometrical constraints is studied. The domain \(\Omega \) is constrained to lie in the half space determined by \(x_{1} \geq 0\) and its boundary to contain a segment of the hyperplane \(\{x_{1} = 0\}\) where non-homogeneous Dirichlet conditions are imposed. We are then looking for the solution of a partial differential equation satisfying a Dirichlet and a Neumann boundary condition simultaneously on the free boundary. The existence and uniqueness of a solution have already been addressed and this paper is devoted first to the study of geometric and asymptotic properties of the solution and then to the numerical treatment of the problem using a shape optimization formulation. The major difficulty and originality of this paper lies in the treatment of the geometric constraints.

MSC:

49J10 Existence theories for free problems in two or more independent variables
35N25 Overdetermined boundary value problems for PDEs and systems of PDEs
35R35 Free boundary problems for PDEs

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References:

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