Abstract
We consider the wave equation with Dirichlet–Neumann boundary conditions on a family of perturbed domains \(\Omega _s\). We discuss the shape differentiability analysis associated with the above mentioned problem, namely the existence of strong material and shape derivatives of the solution, and the rendering of the new wave problem whose solution is given by the shape derivative. The study shows that the Neumann boundary conditions completely change the focus and strategy involved in the shape differentiability analysis, in comparison to the case of the wave equation with purely Dirichlet boundary conditions. In this paper we show that for the existence of weak material derivative, the classical sensitivity analysis of the state can be bypassed by using parameter differentiability of a functional expressed in the form of Min–Max of a convex–concave Lagrangian with saddle point. Then we analyze the strong material derivative via a brute force estimate on the differential quotient, using known regularity results on the solution of the wave problem.
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Bochniak, M.: Linear elliptic boundary value problems in varying domains. Math. Nachr. 250, 17–24 (2003)
Bociu, L., Castle, L., Martin, K., Toundykov, D.: Optimal control in a free boundary fluid-elasticity interaction. In: AIMS Proceedings, pp. 122–131 (2015). doi:10.3934/proc.2015.0122
Bociu, L., Toundykov, D., Zolésio, J.-P.: Well-posedness analysis for the total linearization of a fluid-elasticity interaction. SIAM J. Math. Anal. 47–3, 1958–2000 (2015)
Bociu, L., Zolésio, J.-P.: Strong shape derivative for the wave equation with Neumann boundary condition. In: Homberg, D., Troltzsch, F. (eds.) CSMO 2011, IFIP AICT 391, IFIP International Federation for Information Processing, pp. 445–460 (2013)
Bociu, L., Zolésio, J.-P.: A pseudo-extractor approach to hidden boundary regularity for the wave equation with Neumann boundary conditions. J. Differ. Equ. 259(11), 5688–5708 (2015)
Bucur, D., Buttazzo, G.: Variational Methods in Shape Optimization Problems. Birkhauser, Boston (2005)
Cagnol, J., Eller, M.: Boundary regularity for Maxwell’s equations with applications to shape optimization. J. Differ. Equ. 250(2), 1114–1136 (2011)
Cagnol, J., Zolésio, J.-P.: Shape derivative in the wave equation with Dirichlet boundary conditions. J. Differ. Equ. 158(2), 175210 (1999)
Céa, J.: Conception optimale ou identification de formes. Calcul rapide de la dérivée directionnelle de la fonction cout R.A.I.R.O 20, 371–402 (1986)
Céa, J.: Problems of Shape Optimal Design, Optimization of Distributed Parameter Structures. Sijthoff and Noordhoff, vol. II. Alphen aan den Rijn, Netherlands (1981)
Cuer, M., Zolésio, J.P.: Control of singular problem via differentiation of a min-max. Syst. Control Lett. 11(2), 151–158 (1988)
Delfour, M., Zolésio, J.-P.: Oriented distance function and its evolution equation for initial sets with thin boundary. SIAM J. Control Optim. 42(6), 2286–2304 (2004)
Delfour, M.C., Zolésio, J.-P.: Structure of shape derivatives for non smooth domains. J. Funct. Anal. 104, 1 (1992)
Delfour, M.C., Zolésio, J.-P.: Shape analysis via oriented distance function. J. Funct. Anal. 123, 129 (1994)
Delfour, M.C., Zolésio, J.-P.: Shape sensitivity analysis via min max differentiability. SIAM Control Optim. 26(4), 834–862 (1988)
Delfour, M.C., Zolésio, J.-P.: Further developments in the application of min max differentiability to shape sensitivity analysis. In: Control of Partial Differential Equations. Lecture Notes in Control and Information Sciences, vol. 114, p. 108119. Springer, Berlin (1989)
Delfour, M., Zolésio, J.-P.: Shapes and Geometries. Analysis, Differential Calculus, and Optimization. SIAM Advances in Design and Control (2001)
Delfour, M.C., Zolésio, J.-P.: Hidden boundary smoothness for some classes of differential equations on submanifolds. In: Optimization Methods in Partial Differential Equations. Contemporary Mathematics, vol. 209, p. 5973. American Mathematical Society, Providence, RI (1997)
Delfour, M.C., Zolésio, J.-P.: On the design and control of systems governed by differential equations on sub manifolds. Control Cybern. 25, 497–514 (1996)
Desaint, F.R., Zolésio, J.P.: Manifold derivative in the Laplace–Beltrami equation. J. Funct. Anal. 151(1), 234–269 (1997)
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)
Hettlich, F.: The domain derivative of time-harmonic electromagnetic waves at interfaces. Math. Methods Appl. Sci. 35(14), 1681–1689 (2012)
Ito, K., Kunisch, K., Peichl, G.H.: Variational approach to shape derivatives. ESAIM Control Optim. Calc. Var. 14(3), 517–539 (2008)
Lasiecka, I., Triggiani, R.: Control Theory for Partial Differential Equation. Encyclopedia of Mathematics. Cambridge University Press, Cambridge (2000)
Lasiecka, I., Triggiani, R., Lasiecka, I., Triggiani, R.: Sharp regularity theory for second order hyperbolic equations Neumann type. Part I. \(L_2\) nonhomogeneous data. Ann. Mat. Pura Appl. (IV) CLVII, 285–367 (1990)
Lasiecka, I., Triggiani, R.: Regularity theory of hyperbolic equations with non-homogeneous Neumann boundary conditions. II. General boundary data. JDE 94, 112–164 (1991)
Lions, J.L., Magenes, E.: Nonhomogeneous Boundary Value Problems and Applications. Springer, Berlin (1972)
Sokolowski, J., Zolésio, J.P.: Introduction to shape optimization. Shape sensitivity analysis. Springer Series in Computational Mathematics, vol. 16. Springer, Berlin (1992)
Tataru, D.: On the regularity of boundary traces for the wave equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26(1), 185–206 (1998)
Triggiani, R: Private communication
Zolésio, J.P.: The material derivative (or speed) method for shape optimization. In: Haug, E.J., Céa, J. (eds.) Optimization of Distributed Parameter Structures, vol. II (Iowa City, Iowa, 1980). NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, Sijhofff and Nordhoff, vol. 50, pp. 1089–1151. Alphen aan den Rijn, Nijhoff, The Hague (1981)
Zolésio, J.-P.: Hidden boundary shape derivative for the solution to Maxwell equations and non cylindrical wave equations. In: Optimal Control of Coupled Systems of Partial Differential Equations, International Series of Numerical Mathematics, vol. 158, p. 319345. Birkhuser Verlag, Basel (2009)
Zolésio, J.-P.: Introduction to shape optimization and free boundary problems. In: Shape Optimization and Free Boundaries. Nato Science Series C: Mathematical and Physical Sciences, vol. 380, pp. 397-457. Kluwer Academic, Dordrecht (1992)
Acknowledgments
The authors would like to thank the anonymous referees for their insightful comments and corrections. The research of first author was partially supported by NSF grant DMS-1312801.
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Bociu, L., Zolésio, JP. Hyperbolic Equations with Mixed Boundary Conditions: Shape Differentiability Analysis. Appl Math Optim 76, 375–398 (2017). https://doi.org/10.1007/s00245-016-9354-4
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DOI: https://doi.org/10.1007/s00245-016-9354-4