On the Shape Gradient and Shape Hessian of a Shape Functional Subject to Dirichlet and Robin Conditions

Abstract

This paper focuses on minimizing a shape functional through the solution of a Pure Dirichlet boundary value problem, and a Dirichlet-Robin boundary value problem. This shape optimization problem is a variant of the Kohn-Vogelius shape optimization formulation of a Bernoulli free boundary problem. The fi rst- and second-order shape derivatives of the cost functional under consideration are explicitly derived. Interestingly, the present fi ndings coincide with the existing results regarding solutions to the Bernoulli problem.