The exterior Bernoulli free boundary problem is being considered. The solution to the problem is studied via shape optimization techniques. The goal is to determine a domain having a specific regularity that gives a minimum value for the Kohn-Vogelius-type cost functional while simultaneously solving two PDE constraints: a pure Dirichlet boundary value problem and a Neumann boundary value problem. This paper focuses on the rigorous computation of the first-order shape derivative of the cost functional using the Hölder continuity of the state variables and not the usual approach which uses the shape derivatives of states.
"On the First-Order Shape Derivative of the Kohn-Vogelius Cost Functional of the Bernoulli Problem." Abstr. Appl. Anal. 2013 1 - 19, 2013. https://doi.org/10.1155/2013/384320