Maximum principle in the domain shape optimization in an elliptic problem (Q1333641)

The paper concerns the control problem of maximizing a cost function \(J_ 0(\Omega)\) subject to a finite number of constraints \(J_ i(\Omega)\geq 0\). Here \(\Omega\), the control variable, is a domain in \(\mathbb{R}^ 3\), \(J_ i(\Omega)= \int_ \Omega g_ i(x, u(x)) dx\), and \(u\) is the solution to an elliptic problem: \(Lu= f\) on \(\Omega\) and \(u= \psi\) on \(\partial\Omega\). The admissible controls are specified within the family of domains whose boundaries possess the form \(r= \tau(\varphi)\) in polar coordinates. Given an optimal \(\tau_ 0\), a Hamiltonian is considered and the maximum principle is established.