On Bernoulli's free boundary problem with a random boundary

This article is dedicated to the solution of Bernoulli’s exterior free boundary problem in the situation of a random interior boundary. We describe two different frameworks to define the expectation and the deviation of the resulting annular domain. 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.