Numerical solution of an optimization problem with unknown boundary (Q760179)

Two numerical algorithms are proposed for the following optimization problem: Given \(S_ 0>0\) and a bounded simple closed curve \(\Gamma_ 0\subset R^ 2\), in the family \({\mathcal D}\) of all bounded double connected domains \(\Omega \subset R^ 2\) that have area \(S_ 0\) and have \(\Gamma_ 0\) as interior boundary, find \(\Omega^*\in {\mathcal D}\) that minimizes \(F(\Omega)=\int_{\Omega}| \nabla u_{\Omega}|^ 2dx_ 1dx_ 2\) where \(u_{\Omega}\) is harmonic on \(\Omega\) and satisfies the conditions: \(u_{\Omega}| \Gamma_ 0=1\), \(u_{\Omega}| \Gamma_ c=0\), \(\Gamma_ c\) being the outside boundary of \(\Omega\). The first algorithm, based on some necessary optimality conditions, starts from an admissible \(\Omega_ 0\in {\mathcal D}\) and finds a sequence of admissible domains, \(\Omega_ n\in {\mathcal D}\) for which \(F(\Omega_ n)\) is decreasing; the second algorithm is a variant of the first one in the case \(\Gamma_ 0\) is star-shaped with respect to the origin. Numerical experiments on four examples are presented.