Geodesic
Minimal and maximal solution maps of elliptic QVIs of obstacle type: Lipschitz stability, differentiability, and optimal control
Amal Alphonse1 orcid ; Michael Hintermüller2 ; Carlos N. Rautenberg3 ; Gerd Wachsmuth4 orcid
1Weierstrass Institute, Berlin, Germany
2Weierstrass Institute, Berlin, Germany; Humboldt-Universität zu Berlin, Germany
3George Mason University, Fairfax, USA
4Brandenburgische Technische Universität Cottbus-Senftenberg, Germany
Interfaces and free boundaries, Tome 27 (2025) no. 4, pp. 521-573

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DOI

Quasi-variational inequalities (QVIs) of obstacle type in many cases have multiple solutions that can be ordered. We study a multitude of properties of the operator mapping the source term to the minimal or maximal solution of such QVIs. We prove that the solution maps are locally Lipschitz continuous and directionally differentiable and show existence of optimal controls for problems that incorporate these maps as the control-to-state operator. We also consider a Moreau–Yosida-type penalisation for the QVI, wherein we show that it is possible to approximate the minimal and maximal solutions by sequences of minimal and maximal solutions (respectively) of certain PDEs, which have a simpler structure and offer a convenient characterisation in particular for computation. For solution mappings of these penalised problems, we prove a number of properties including Lipschitz and differential stability. Making use of the penalised equations, we derive (in the limit) C-stationarity conditions for the control problem, in addition to the Bouligand stationarity we get from the differentiability result.
DOI : 10.4171/ifb/545
Classification : 49J40, 47J20, 49J52, 49J21, 49K21
Mots-clés : quasi-variational inequality, optimal control, stationary system, ordered solution, directional differentiability

Amal Alphonse  1   ; Michael Hintermüller  2   ; Carlos N. Rautenberg  3   ; Gerd Wachsmuth  4

1 Weierstrass Institute, Berlin, Germany
2 Weierstrass Institute, Berlin, Germany; Humboldt-Universität zu Berlin, Germany
3 George Mason University, Fairfax, USA
4 Brandenburgische Technische Universität Cottbus-Senftenberg, Germany 
Amal Alphonse; Michael Hintermüller; Carlos N. Rautenberg; Gerd Wachsmuth. Minimal and maximal solution maps of elliptic QVIs of obstacle type: Lipschitz stability, differentiability, and optimal control. Interfaces and free boundaries, Tome 27 (2025) no. 4, pp. 521-573. doi: 10.4171/ifb/545
@article{10_4171_ifb_545,
     author = {Amal Alphonse and Michael Hinterm\"uller and Carlos N. Rautenberg and Gerd Wachsmuth},
     title = {Minimal and maximal solution maps of elliptic {QVIs} of obstacle type: {Lipschitz} stability, differentiability, and optimal control},
     journal = {Interfaces and free boundaries},
     pages = {521--573},
     year = {2025},
     volume = {27},
     number = {4},
     doi = {10.4171/ifb/545},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/545/}
}
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