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International Journal for Uncertainty Quantification
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ISSN Imprimer: 2152-5080
ISSN En ligne: 2152-5099

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International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2017019550
pages 335-353

ON BERNOULLI'S FREE BOUNDARY PROBLEM WITH A RANDOM BOUNDARY

M. Dambrine
Université de Pau et des Pays de l'Adour, IPRA-LMA, UMR CNRS 5142 Avenue de l'université, 64000 Pau, France
Helmut Harbrecht
Universität Basel, Departement Mathematik und Informatik, Spiegelgasse 1, 4051 Basel, Switzerland
M. D. Peters
Universität Basel, Departement Mathematik und Informatik, Spiegelgasse 1, 4051 Basel, Switzerland
B. Puig
Université de Pau et des Pays de l'Adour, IPRA-LMA, UMR CNRS 5142 Avenue de l'université, 64000 Pau, France

RÉSUMÉ

This article is dedicated to the solution of Bernoulli's exterior free boundary problem in the situation of a random interior boundary. We provide the theoretical background that ensures the well-posedness of the problem under consideration and describe two different frameworks to define the expectation and the deviation of the resulting annular domain. The first approach is based on the Vorob'ev expectation, which can be defined for arbitrary sets. The second approach is based on the particular parametrization. In order to compare these approaches, we present analytical examples for the case of a circular interior boundary. Additionally, numerical experiments are performed for more general geometric configurations. For the numerical approximation of the expectation and the deviation, we propose a sampling method like the Monte Carlo or the quasi-Monte Carlo quadrature. Each particular realization of the free boundary is then computed by the trial method, which is a fixed-point-like iteration for the solution of Bernoulli's free boundary problem.


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