International Journal of Differential Equations

Volume 2015 (2015), Article ID 954836, 10 pages

http://dx.doi.org/10.1155/2015/954836

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

Department of Mathematics and Computer Science, College of Science, University of the Philippines Baguio, Governor Pack Road, 2600 Baguio City, Philippines

Received 31 July 2015; Revised 11 November 2015; Accepted 11 November 2015

Academic Editor: Julio D. Rossi

Copyright © 2015 Jerico B. Bacani and Julius Fergy T. Rabago. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The exterior Bernoulli free boundary problem was studied via shape optimization technique. The problem was reformulated into the minimization of the so-called Kohn-Vogelius objective functional, where two state variables involved satisfy two boundary value problems, separately. The paper focused on solving the second-order shape derivative of the objective functional using the velocity method with nonautonomous velocity fields. This work confirms the classical results of Delfour and Zolésio in relating shape derivatives of functionals using velocity method and perturbation of identity technique.

#### 1. Introduction

Shape optimization is a key research topic with many applications in various fields of pure and applied sciences, especially in biomechanics and engineering (cf. [1, 2] for applications in structural mechanics, [3] for some applications in fluid mechanics or aerodynamics, and [4] for other applications). A typical problem in this line of research is to find a domain, for instance, , in a set of admissible domains such that an objective functional achieves a minimum (or maximum) on it [3]. For instance, suppose, among all three-dimensional shapes of given volume, that we wish to find the one which has a minimal surface area. In this particular case, the problem can be described mathematically as finding the minimum of with the constraint . Obviously, the answer to this question would be the sphere. In general and in most cases of greater interest, shape optimization problems can be described mathematically aswhere the state is the solution to a partial differential equation (PDE) on the domain . For an extensive introduction to shape optimization problems, we refer to the book of Delfour and Zolésio [5] (see also [6]).

Recently, there has been an increasing interest in the applications of shape optimization in the study of Bernoulli problems. Abda et al. [7] rephrased the Bernoulli problem into a shape optimization problem and explicitly determined the shape derivative of the cost functional being studied. In [8], a framework for calculating the* shape Hessian* for the domain optimization problem with a PDE as the constraint was presented. In [9], a similar approach as in [8] was applied in solving a shape optimization problem.

Another way to approach the solutions of shape optimization problems is through iterative methods. For the past few decades, several numerical methods have been developed to solve the two-dimensional Bernoulli problem (see, e.g., [10–13]). These strategies were also developed based on reformulating the Bernoulli problem as a shape optimization problem. This reformulation can be achieved in several ways. For instance, for a given domain, one can choose one of the boundary conditions on the free boundary to obtain a well-posed state equation. The domain is determined by the requirement that the other condition on the free boundary is satisfied in a least square sense (see [13–15]).

Many authors have also studied the second variation of a cost functional for linear PDEs. Building on the shape optimization setting that is based on the* perturbation of the identity method* introduced by Murat and Simon (cf. [16, 17]), Fujii [18] used a second-order perturbation of the identity along the normal of the boundary for second-order elliptic problems in 1986. Simon [19] computed the second variation via the first-order perturbation of the identity in 1988. A general approach via the velocity method (Figure 2) was systematically characterized by Delfour and Zolésio [20, 21], and they computed the shape Hessian for a simple Neumann problem in [20] and a nonhomogeneous Dirichlet problem in [21].

However, a standard approach in dealing with the solution to (1) requires some information on gradients. So shape derivatives are essential in understanding the problem.

The recent paper focuses on the exterior Bernoulli free boundary problem (FBP). 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. As a main result, by analyzing the shape Hessian at the optimal domain, Eppler and Harbrecht found out that the optimization problem is algebraically ill posed. 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.

Now, the exterior Bernoulli FBP is formulated as follows.

Given a bounded and connected domain with a fixed boundary , we need to find a bounded connected domain with a free boundary that contains the closure of , , and an associated real-valued (state) function defined on (where is the annulus formed by and ; refer to Figure 1) such that both unknowns and satisfy the following boundary value problem:where .