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.
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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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Rauls, A., Wachsmuth, G. Generalized Derivatives for the Solution Operator of the Obstacle Problem. Set-Valued Var. Anal 28, 259–285 (2020). https://doi.org/10.1007/s11228-019-0506-y
- Obstacle problem
- Generalized derivative
- Capacitary measure
- Relaxed Dirichlet problem
Mathematics Subject Classification (2010)