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Shape derivative in the wave equation with Dirichlet boundary conditions. (English) Zbl 0959.35017

The authors study the second-order Dirichlet hyperbolic problem \[ \begin{aligned} & \partial^2_ty-\text{div}(K\nabla y)=f\quad \text{on } (0,T)\times \Omega,\\ & y=g\quad \text{on }(0,T)\times \partial\Omega,\\ & y(0)=\varphi,\quad \partial_ty(0)=\psi\quad\text{on }\Omega,\end{aligned} \] where \(K\) be a coercive and symmetric \(N\times N\)-matrix of functions, and \(\Omega\subset\mathbb{R}^n\) is a a domain with the boundary \(\partial\Omega\in C^k\) for \(k \geq 2\). First of all the authors prove that the solution of the given problem is shape and material differentiable under a strong regularity of the data. Then using the hidden regularity they prove the existence of a shape derivative under weak regularity conditions of the data.
The implicit function theorem used in the smooth cases does not work to solve the problem of a weak regularity of the data.
Nevertheless by a more technical approach the authors prove analogical results as for the regular data.

MSC:

35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
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[1] J. Cagnol, and, J.-P. Zolésio, Hidden shape derivative in the wave equation, in, Proceedings of the 18th IFIP TC7 Conference on System Modelling and Optimization, (, M. Polis and I. Lasiecka, Eds.), Addison-Wesley/Longman, Reading, MA/Harlow, in press.; J. Cagnol, and, J.-P. Zolésio, Hidden shape derivative in the wave equation, in, Proceedings of the 18th IFIP TC7 Conference on System Modelling and Optimization, (, M. Polis and I. Lasiecka, Eds.), Addison-Wesley/Longman, Reading, MA/Harlow, in press.
[2] Cagnol, J.; Zolésio, J.-P., Hidden shape derivative in the wave equation with dirichlet boundary condition, C. R. Acad. Sci. Paris Sér. I, 326, 1079-1084 (1998) · Zbl 0935.35011
[3] Delfour, M. C.; Zolésio, J.-P., Structure of shape derivatives for non smooth domains, J. Funct. Anal., 104 (1992) · Zbl 0777.49030
[4] Delfour, M. C.; Zolésio, J.-P., Shape analysis via oriented distance functions, J. Funct. Anal., 123 (1994) · Zbl 0814.49032
[5] Delfour, M. C.; Zolésio, J.-P., Hidden boundary smoothness for some classes of differential equations on submanifolds, (Lasiecka, I.; Cox, S., Joint Summer Research Conference on Optimization Methods in PDE’s. Joint Summer Research Conference on Optimization Methods in PDE’s, Contemporary Mathematics, 209 (1997), Amer. Math. Soc: Amer. Math. Soc Providence), 59 · Zbl 0903.35011
[6] Desaint, F. R.; Zolésio, J.-P., Manifold derivative in the Laplace-Beltrami equation, J. Funct. Anal., 151, 234 (1997) · Zbl 0903.58059
[7] Lasiecka, I.; Lions, J.-L.; Triggiani, R., Non homogeneous boundary boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986) · Zbl 0631.35051
[8] Lasiecka, I.; Triggiani, R., Recent advances in regularity of second-order hyperbolic mixed problems, and applications, Dynamics Reported: Expositions in Dynamical Systems (New Series) (1994), Springer-Verlag: Springer-Verlag New York · Zbl 0807.35080
[9] Lions, J. L.; Magenes, E., Non-homogeneous Boundary Value Problems and Applications (1972), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0223.35039
[10] Sokolowski, J.; Zolésio, J.-P., Introduction to Shape Optimization (1991), Springer-Verlag: Springer-Verlag New York/Berlin
[11] Zolésio, J.-P., Introduction to shape optimization and free boundary problems, (Delfour, M. C., Shape Optimization and Free Boundarys. Shape Optimization and Free Boundarys, NATO ASI Ser. C: Math. Phys. Sci., 380 (1992), Kluwer Academic: Kluwer Academic Dordrecht), 397-457 · Zbl 0765.76070
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