Generalized Derivatives for the Solution Operator of the Obstacle Problem
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We characterize generalized derivatives of the solution operator of the obstacle problem. This precise characterization requires the usage of the theory of so-called capacitary measures and the associated solution operators of relaxed Dirichlet problems. The generalized derivatives can be used to obtain a novel necessary optimality condition for the optimal control of the obstacle problem with control constraints. A comparison shows that this system is stronger than the known system of C-stationarity.
KeywordsObstacle problem Generalized derivative Capacitary measure Relaxed Dirichlet problem C-stationarity
Mathematics Subject Classification (2010)49K40 47J20 49J52 58C20
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The authors would like to thank Giuseppe Buttazzo for pointing out the result of [30, Lemma 4.12] which led to the discovery of Theorem 3.16. Further, Constantin Christof pointed out the result of Theorem 3.9 (f-supp(1 + Δw) =qΩ ∖ O for w = LO(1)) and this is gratefully acknowledged. He gave a different, interesting proof based on differentiability properties of the obstacle problem and this proof might appear elsewhere.
This work is supported by DFG grants UL158/10-1 and WA3636/4-1 within the Priority Program SPP 1962 (Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization).
- 4.Rauls, A.-T., Ulbrich, S.: Subgradient computation for the solution operator of the obstacle problem. Tech. rep., preprint SPP1962-056. https://spp1962.wias-berlin.de/preprints/056.pdf (2018)
- 7.Attouch, H., Buttazzo, G., Michaille, G.: Variational analysis in Sobolev and BV spaces, 2nd edn. Volume 17 of MOS-SIAM Series on Optimization, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Optimization Society, Philadelphia, PA, Applications to PDEs and optimization. https://doi.org/10.1137/1.9781611973488 (2014)
- 12.Rudin, W.: Real and Complex Analysis. Mathematics Series. McGraw-Hill (1987)Google Scholar
- 17.Dal Maso, G., Murat, F.: Asymptotic behaviour and correctors for linear dirichlet problems with simultaneously varying operators and domains, Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 21 (4), 445–486 (2004). https://doi.org/10.1016/j.anihpc.2003.05.001 MathSciNetCrossRefzbMATHGoogle Scholar
- 19.Dal Maso, G.: Γ-convergence and μ-capacities. Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV 14(3), 423–464 (1988) (1987). http://www.numdam.org/item?id=ASNSP_1987_4_14_3_423_0 MathSciNetzbMATHGoogle Scholar
- 23.Velichkov, B.: Existence and Regularity Results for Some Shape Optimization Problems, vol. 19 of Tesi. Scuola Normale Superiore di Pisa (Nuova Series) [Theses of Scuola Normale Superiore di Pisa (New Series)]. Pisa, Edizioni della Normale (2015)Google Scholar
- 28.Cioranescu, D., Murat, F.: A strange term coming from nowhere. In: Topics in the Mathematical Modelling of Composite Materials, vol. 31 of Progr. Nonlinear Differential Equations Appl., pp 45–93. Birkhäuser, Boston (1997)Google Scholar