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bernoulli condition

ESAIM: Control, Optimisation and Calculus of Variations (1)

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On a Bernoulli problem with geometric constraints
Antoine Laurain, Yannick Privat
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 18 / Issue 1 / January 2012

Published online by Cambridge University Press:

02 December 2010, pp. 157-180

Print publication:

January 2012
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A Bernoulli free boundary problem with geometrical constraints is studied. The domain Ω is constrained to lie in the half space determined by x1 ≥ 0 and its boundary to contain a segment of the hyperplane {x1 = 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.