Adjoint methods for obstacle problems and weakly coupled systems of PDE
ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 3, pp. 754-779.

The adjoint method, recently introduced by Evans, is used to study obstacle problems, weakly coupled systems, cell problems for weakly coupled systems of Hamilton - Jacobi equations, and weakly coupled systems of obstacle type. In particular, new results about the speed of convergence of some approximation procedures are derived.

DOI : https://doi.org/10.1051/cocv/2012032
Classification : 35F20,  35F30,  37J50,  49L25
Mots clés : adjoint methods, cell problems, Hamilton − Jacobi equations, obstacle problems, weakly coupled systems, weak KAM theory
@article{COCV_2013__19_3_754_0,
     author = {Cagnetti, Filippo and Gomes, Diogo and Tran, Hung Vinh},
     title = {Adjoint methods for obstacle problems and weakly coupled systems of {PDE}},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {754--779},
     publisher = {EDP-Sciences},
     volume = {19},
     number = {3},
     year = {2013},
     doi = {10.1051/cocv/2012032},
     zbl = {1273.35090},
     mrnumber = {3092361},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2012032/}
}
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Cagnetti, Filippo; Gomes, Diogo; Tran, Hung Vinh. Adjoint methods for obstacle problems and weakly coupled systems of PDE. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 3, pp. 754-779. doi : 10.1051/cocv/2012032. http://www.numdam.org/articles/10.1051/cocv/2012032/

[1] G. Barles and B. Perthame, Exit time problems in optimal control and vanishing viscosity method. SIAM J. Control Optim. 26 (1988) 1133-1148. | MR 957658 | Zbl 0674.49027

[2] I. Capuzzo-Dolcetta and L.C. Evans, Optimal switching for ordinary differential equations. SIAM J. Control Optim. 22 (1984) 143-161. | MR 728678 | Zbl 0641.49017

[3] F. Cagnetti, D. Gomes and H.V. Tran, Aubry-Mather measures in the nonconvex setting. SIAM J. Math. Anal. 43 (2011) 2601-2629. | MR 2873233 | Zbl 1258.35059

[4] F. Camilli and P. Loreti, Comparison results for a class of weakly coupled systems of eikonal equations. Hokkaido Math. J. 37 (2008) 349-362. | MR 2415905

[5] F. Camilli, P. Loreti, and N. Yamada, Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem. Commun. Pure Appl. Anal. 8 (2009) 1291-1302. | MR 2505371 | Zbl 1160.49028

[6] H. Engler and S.M. Lenhart, Viscosity solutions for weakly coupled systems of Hamilton-Jacobi equations. Proc. London Math. Soc. 63 (1991) 212-240. | MR 1105722 | Zbl 0704.35030

[7] L.C. Evans and C.K. Smart, Adjoint methods for the infinity Laplacian partial differential equation. Arch. Ration. Mech. Anal. 201 (2011) 87-113. | MR 2807134 | Zbl 1257.35089

[8] L.C. Evans, Adjoint and compensated compactness methods for Hamilton-Jacobi PDE. Arch. Ration. Mech. Anal. 197 (2010) 1053-1088. | MR 2679366 | Zbl 1273.70030

[9] D.A. Gomes, A stochastic analogue of Aubry-Mather theory. Nonlinearity 15 (2002) 581-603. | MR 1901094 | Zbl 1073.37078

[10] H. Ishii and S. Koike, Viscosity solutions for monotone systems of second-order elliptic PDEs. Commun. Partial Differ. Equ. 16 (1991) 1095-1128. | MR 1116855 | Zbl 0742.35022

[11] K. Ishii and N. Yamada, On the rate of convergence of solutions for the singular perturbations of gradient obstacle problems. Funkcial. Ekvac. 33 (1990) 551-562. | MR 1086777 | Zbl 0728.35006

[12] P.L. Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Mathematics. Pitman (Advanced Publishing Program), Boston, Mass. 69 (1982). | MR 667669 | Zbl 0497.35001

[13] P.L. Lions, G. Papanicolaou and S.R.S. Varadhan, Homogenization of Hamilton-Jacobi equations, Preliminary Version, (1988).

[14] H.V. Tran, Adjoint methods for static Hamilton-Jacobi equations. Calc. Var. Partial Differ. Equ. 41 (2011) 301-319. | MR 2796233 | Zbl 1231.35043

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