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).
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