Abstract
In this paper we consider a special type of optimization problems for elliptic, parabolic and hyperbolic partial differential equations, in which the space domains play the role of the controls. It is demonstrated that some of the free boundary problems may be formulated as shape optimization problems. To solve them, it is suggested to use the shape penalty method. It is also shown that the solution of one-dimensional and one-phase Stefan problems is a limit of the solutions of optimization problems for standard parabolic systems in a fixed domain. We prove theorems concerning the convergence of the approximate systems to the solution of the Stefan problem and present some numerical illustrations.
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