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International Journal of Differential Equations
Volume 2015, Article ID 954836, 10 pages
Research Article

On the Second-Order Shape Derivative of the Kohn-Vogelius Objective Functional Using the Velocity Method

Department of Mathematics and Computer Science, College of Science, University of the Philippines Baguio, Governor Pack Road, 2600 Baguio City, Philippines

Received 31 July 2015; Revised 11 November 2015; Accepted 11 November 2015

Academic Editor: Julio D. Rossi

Copyright © 2015 Jerico B. Bacani and Julius Fergy T. Rabago. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The exterior Bernoulli free boundary problem was studied via shape optimization technique. The problem was reformulated into the minimization of the so-called Kohn-Vogelius objective functional, where two state variables involved satisfy two boundary value problems, separately. The paper focused on solving the second-order shape derivative of the objective functional using the velocity method with nonautonomous velocity fields. This work confirms the classical results of Delfour and Zolésio in relating shape derivatives of functionals using velocity method and perturbation of identity technique.