Generalized Derivatives for the Solution Operator of the Obstacle Problem

Abstract

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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Acknowledgments

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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Correspondence to Gerd Wachsmuth.

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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

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Keywords

  • Obstacle problem
  • Generalized derivative
  • Capacitary measure
  • Relaxed Dirichlet problem
  • C-stationarity

Mathematics Subject Classification (2010)

  • 49K40
  • 47J20
  • 49J52
  • 58C20