On a Kohn-Vogelius like formulation of free boundary problems

Abstract

The present paper is concerned with the solution of a Bernoulli type free boundary problem by means of shape optimization. Two state functions are introduced, namely one which satisfies the mixed boundary value problem, whereas the second one satisfies the pure Dirichlet problem. The shape problem under consideration is the minimization of the L 2 -distance of the gradients of the state functions. We compute the corresponding shape gradient and Hessian. By the investigation of sufficient second order conditions we prove algebraic ill-posedness of the present formulation. Our theoretical findings are supported by numerical experiments.

On the Second-Order Shape Derivative of the Kohn-Vogelius Objective Functional Using the Velocity Method

“…We remark that, as compared to the form of the shape Hessian presented in [22], the result we have established here in Theorem 9 clearly shows the relation pointed in [26] by Delfour and Zolésio about the form of shape Hessians obtained through nonautonomous velocity fields.…”

“…The above result was first proven by Eppler and Harbrecht in [22]. Other proofs were also given by Bacani and Peichl [9,23], by using two different approaches.…”

“…As far as the authors are concerned, the same functional was first studied by Eppler and Harbrecht and published in [22] wherein the first-order shape derivative, or equivalently the shape gradient, was derived for arbitrary variations in terms of the perturbation of the identity. Moreover, the second-order shape derivative, or equivalently the shape Hessian, has been computed and analyzed for the special cases of star-like domains.…”

“…In the present paper, the same functional is studied again but we focus on the application of velocity method in dealing with shape optimization problem. It would be a challenging research in the near future to study the ill-posedness of the shape optimization problem for general domains, as well as the comparison of the shape Hessians in this paper from [22] for the former uses Cartesian coordinates, while the latter used spherical/polar coordinates. The nice thing in the present paper is that the results attest to classical results in shape optimization problems.…”

A free boundary problem for the Stokes equations

“…Assuming the existence of a solution to (1.1) we adopt the so-called KohnVogelius formulation, which applies for inverse problems (see [13,14,26] for instance) and which consists in matching a Dirichlet and a Neumann problem. For an application of this technique to the Bernoulli free boundary problem we refer to [3,9].…”