Optimal regularity in the optimal switching problem

Aleksanyan, Gohar

doi:10.1016/j.anihpc.2015.06.001

Aleksanyan, Gohar

Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 6, pp. 1455-1471.

Résumé

In this article we study the optimal regularity for solutions to the following weakly coupled system with interconnected obstacles

{\begin{matrix} \min (- Δ u^{1} + f^{1}, u^{1} - u^{2} + ψ^{1}) = 0 \\ \min (- Δ u^{2} + f^{2}, u^{2} - u^{1} + ψ^{2}) = 0, \end{matrix}

arising in the optimal switching problem with two modes.

We derive the optimal $C^{1, 1}$ -regularity for the minimal solution under the assumption that the zero loop set $L : = {ψ^{1} + ψ^{2} = 0}$ is the closure of its interior. This result is optimal and we provide a counterexample showing that the $C^{1, 1}$ -regularity does not hold without the assumption $L = \overline{L^{0}}$ .

Zbl

DOI : 10.1016/j.anihpc.2015.06.001

Mots clés : Optimal switching problem, Regularity theory, The obstacle problem, Double obstacle problem, Free boundary problem, Nonlinear elliptic system

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     author = {Aleksanyan, Gohar},
     title = {Optimal regularity in the optimal switching problem},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1455--1471},
     publisher = {Elsevier},
     volume = {33},
     number = {6},
     year = {2016},
     doi = {10.1016/j.anihpc.2015.06.001},
     zbl = {1352.49036},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2015.06.001/}
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Aleksanyan, Gohar. Optimal regularity in the optimal switching problem. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 6, pp. 1455-1471. doi : 10.1016/j.anihpc.2015.06.001. http://archive.numdam.org/articles/10.1016/j.anihpc.2015.06.001/

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