Shape optimization for free boundary problems

In this paper three different formulations of a Bernoulli type free boundary problem are discussed. By analyzing the shape Hessian in case of matching data it is distinguished between well-posed and ill-posed formulations. A nonlinear Ritz-Galerkin method is applied for discretizing the shape optimization problem. In case of well-posedness existence and convergence of the approximate shapes is proven. In combination with a fast boundary element method efficient first and second order shape optimization algorithms are obtained. 1 Problem formulation The present paper is dedicated to the solution of a generalized Bernoulli exterior free boundary problem which serves as a prototype of many shape optimization problems. Let T ⊂ R denote a bounded domain with free boundary ∂T = Γ. Inside the domain T we assume the existence of a simply connected subdomain S ⊂ T with fixed boundary ∂S = Σ. The resulting annular domain T \ S is denoted by Ω. The exterior free boundary problem might be formulated as follows: For given data f, g, h, seek the domain Ω and associated function u such that the overdetermined boundary value problem −∆u = f in Ω, − ∂n = g, u = 0 on Γ, u = h on Σ, (1.1) is satisfied. Here, g, h > 0 and f ≥ 0 are sufficiently smooth functions on R such that u provides enough regularity for a second order shape calculus. We like to stress that the positivity of the Dirichlet data implies that u is positive on Ω and thus it holds in fact ∂u/∂n < 0. We are going to consider the following formulations: (i) If we prescribe the Dirichlet data u = 0 at the free boundary problem, then the solution of (1.1) is the minimizer of the Dirichlet energy functional (cf. [4])