Optimal regularity in the optimal switching problem
Annales de l'I.H.P. Analyse non linéaire, Volume 33 (2016) no. 6, pp. 1455-1471.

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

{min(Δu1+f1,u1u2+ψ1)=0min(Δu2+f2,u2u1+ψ2)=0,
arising in the optimal switching problem with two modes.

We derive the optimal C1,1-regularity for the minimal solution under the assumption that the zero loop set L:={ψ1+ψ2=0} is the closure of its interior. This result is optimal and we provide a counterexample showing that the C1,1-regularity does not hold without the assumption L=L0.

DOI: 10.1016/j.anihpc.2015.06.001
Keywords: Optimal switching problem, Regularity theory, The obstacle problem, Double obstacle problem, Free boundary problem, Nonlinear elliptic system
@article{AIHPC_2016__33_6_1455_0,
     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://www.numdam.org/articles/10.1016/j.anihpc.2015.06.001/}
}
TY  - JOUR
AU  - Aleksanyan, Gohar
TI  - Optimal regularity in the optimal switching problem
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2016
SP  - 1455
EP  - 1471
VL  - 33
IS  - 6
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2015.06.001/
DO  - 10.1016/j.anihpc.2015.06.001
LA  - en
ID  - AIHPC_2016__33_6_1455_0
ER  - 
%0 Journal Article
%A Aleksanyan, Gohar
%T Optimal regularity in the optimal switching problem
%J Annales de l'I.H.P. Analyse non linéaire
%D 2016
%P 1455-1471
%V 33
%N 6
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2015.06.001/
%R 10.1016/j.anihpc.2015.06.001
%G en
%F AIHPC_2016__33_6_1455_0
Aleksanyan, Gohar. Optimal regularity in the optimal switching problem. Annales de l'I.H.P. Analyse non linéaire, Volume 33 (2016) no. 6, pp. 1455-1471. doi : 10.1016/j.anihpc.2015.06.001. http://www.numdam.org/articles/10.1016/j.anihpc.2015.06.001/

[1] Andersson, John; Lindgren, Erik; Shahgholian, Henrik Optimal regularity for the no-sign obstacle problem, Commun. Pure Appl. Math., Volume 66 (2013), pp. 245–262 | Zbl

[2] Belbas, Stavros A.; Lenhart, Suzanne M. Nonlinear PDEs for stochastic optimal control with switchings and impulses, Appl. Math. Optim., Volume 14 (1986), pp. 215–227

[3] Cagnetti, Filippo; Gomes, Diogo; Tran, Hung Vinh Adjoint methods for obstacle problems and weakly coupled systems of PDE, ESAIM Control Optim. Calc. Var., Volume 19 (2013), pp. 754–779 | Numdam | Zbl

[4] Evans, Lawrence C.; Friedman, Avner Optimal stochastic switching and the Dirichlet problem for the Bellman equation, Trans. Am. Math. Soc., Volume 253 (1979), pp. 365–389 | Zbl

[5] Gilbarg, David; Trudinger, Neil S. Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001 (reprint of the 1998 edition) | DOI | Zbl

[6] Gomes, Diogo; Serra, António Systems of weakly coupled Hamilton–Jacobi equations with implicit obstacles, Can. J. Math., Volume 64 (2012), pp. 1289–1309 | Zbl

[7] Lenhart, Suzanne M.; Belbas, Stavros A. A system of nonlinear partial differential equations arising in the optimal control of stochastic systems with switching costs, SIAM J. Appl. Math., Volume 43 (1983), pp. 465–475 | Zbl

[8] Petrosyan, Arshak; Shahgholian, Henrik; Uraltseva, Nina Regularity of Free Boundaries in Obstacle-Type Problems, American Mathematical Society, Providence, RI, 2012 | DOI | Zbl

[9] Stein, Elias M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993 | Zbl

Cited by Sources: