Generalized Derivatives for the Solution Operator of the Obstacle Problem

  • Anne-Therese Rauls
  • Gerd WachsmuthEmail author


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.


Obstacle 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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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Research Group Optimization, Department of MathematicsTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Chair of Optimal Control, Institute of MathematicsBrandenburgische Technische Universität Cottbus-SenftenbergCottbusGermany

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