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Variational approach to shape derivatives for a class of Bernoulli problems. (English) Zbl 1088.49028

The authors consider the shape optimization problem \(\inf_{\Gamma }\int_{\Gamma }u^{2}d\Gamma /2\), where \(u\) is the solution of the elliptic problem: \(-\Delta u=f\) in \(\Omega \), with mixed boundary conditions: \(u=u_{d} \) on \(\Gamma _{d}\), \(\partial u/\partial n=g\) on \(\Gamma \). Here \(\Omega \) is a smooth and bounded domain of \(\mathbb{R}^{2}\), the boundary of which is decomposed in the disjoint union of a fixed part \(\Gamma _{d}\) and \(\Gamma \), both with nonempty relative interior. The authors assume that \(\Omega \) is contained in a fixed convex and open subset \(U\) of \(\mathbb{R}^{2}\). In the above elliptic problem, \(f\in H^{s}\left( U\right) \), \(s>1/2\), \(u_{d}\in H^{3/2}\left( \Gamma _{d}\right) \), \(g\in H^{2}\left( U\right) \). The purpose of the paper is to compute the shape derivative of the above cost functional, without computing the shape derivative of the state, as usually done. This kind of problem occurs in free boundary problems of Bernoulli type. For the computation of the shape derivative, the authors introduce perturbations. The perturbed domain \(\Omega _{t}\) is obtained through \( \Omega _{t}=F_{t}\left( \Omega \right) \), where \(F_{t}\) is the transformation given on \(\Omega \) through \(F_{t}\left( x\right) =x+th\left( x\right) \), \(h\) being taken in the class \(\left\{ h\in \left( C^{1,1}\left( \overline{U}\right) \right) ^{2}\mid h=0\text{ on }\Gamma _{d}\right\} \). The main result of the paper expresses this shape derivative in terms of the state, of the data and of the mean curvature of \(\Gamma \). Using this expression of the shape derivative and a level set approach, the authors then build a numerical scheme which solves this problem. This scheme mainly consists to write the perturbed boundary \(\Gamma _{t}\) as the zero level set of some function \(\Psi \), then \(\Omega _{t}\) as the set \(\left\{ \Psi <0\right\} \). This leads to a Hamilton-Jacobi problem satisfied by \(\Psi \). The algorithm uses this Hamilton-Jacobi equation in order to compute the perturbation \(h\), through the shape derivative of the cost functional, and then the updated perturbed domain \(\Omega _{t}\). The paper ends with some numerical computations, first in the case where \(\Gamma _{d}\) is a circle, then for an L-shape domain and the Bernoulli problem.

MSC:

49Q10 Optimization of shapes other than minimal surfaces
35A15 Variational methods applied to PDEs
35J60 Nonlinear elliptic equations
35R35 Free boundary problems for PDEs
49M25 Discrete approximations in optimal control
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