In this paper we consider a model shape optimization problem. The state variable solves an elliptic equation on a domain with one part of the boundary described as the graph of a control function. We prove higher regularity of the control and develop a priori error analysis for the finite element discretization of the shape optimization problem under consideration. The derived a priori error estimates are illustrated on two numerical examples.

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In this paper we consider a model shape optimization problem. The state variable solves an elliptic equation on a star-shaped domain where the radius is given via a control function. First we reformulate the problem on a fixed reference domain, where we put a focus on the regularity which is needed to ensure the existence of an optimal solution. Second, we introduce the Lagrangian and use it to show that the optimal solution possesses a higher regularity, which allows for the explicit computation of the derivative of the reduced cost functional as a boundary integral. We finish the paper with some second order optimality conditions.