Solving the Exterior Bernoulli Problem Using the Shape Derivative Approach

In this paper, we are interested in solving the exterior Bernoulli free boundary problem by minimizing a particular cost functional \(J\) over a class of admissible domains subject to two well-posed PDE constraints: a Dirichlet boundary value problem and a Neumann boundary value problem. The main result for this paper is the thorough computation of the first-order shape derivative of \(J\) using the shape derivatives of the state variables. At first, the material derivatives of the states are rigorously justified. Then the equation and the boundary conditions satisfied by the corresponding shape derivatives are derived directly from the definition of the shape derivative and the variational equation for the material derivative. It becomes apparent that the analysis of the shape derivatives of the states requires more regular domains. Finally, it is noted that the shape gradient agrees with the structure predicted by the Hadamard structure theorem.