Variational approach to shape derivatives

¹ Department of Mathematics, North Carolina State University, Raleigh, North Carolina, USA; kito@unity.ncsu.edu
² Institute for Mathematics and Scientific Computing, Karl-Franzens-University Graz, 8010 Graz, Austria; karl.kunisch@uni-graz.at; gunther.peichl@uni-graz.at

Received: 27 June 2006
Revised: 19 December 2006

Abstract

A general framework for calculating shape derivatives for optimization problems with partial differential equations as constraints is presented. The proposed technique allows to obtain the shape derivative of the cost without the necessity to involve the shape derivative of the state variable. In fact, the state variable is only required to be Lipschitz continuous with respect to the geometry perturbations. Applications to inverse interface problems, and shape optimization for elliptic systems and the Navier-Stokes equations are given.