On a Bernoulli problem with geometric constraints

TL;DR

This paper investigates a Bernoulli free boundary problem with geometric constraints, analyzing the solution's properties and developing a numerical approach via shape optimization, addressing challenges posed by the geometric restrictions.

Contribution

It introduces a novel analysis of geometric and asymptotic properties of solutions and proposes a shape optimization method for numerical treatment under geometric constraints.

Findings

Characterization of geometric properties of solutions

Asymptotic behavior analysis

Development of a shape optimization numerical method

Abstract

A Bernoulli free boundary problem with geometrical constraints is studied. The domain $\Om$ 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.