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 : 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},
     mrnumber = {3092361},
     zbl = {1273.35090},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2012032/}
}
TY  - JOUR
AU  - Cagnetti, Filippo
AU  - Gomes, Diogo
AU  - Tran, Hung Vinh
TI  - Adjoint methods for obstacle problems and weakly coupled systems of PDE
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2013
SP  - 754
EP  - 779
VL  - 19
IS  - 3
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2012032/
DO  - 10.1051/cocv/2012032
LA  - en
ID  - COCV_2013__19_3_754_0
ER  - 
%0 Journal Article
%A Cagnetti, Filippo
%A Gomes, Diogo
%A Tran, Hung Vinh
%T Adjoint methods for obstacle problems and weakly coupled systems of PDE
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2013
%P 754-779
%V 19
%N 3
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2012032/
%R 10.1051/cocv/2012032
%G en
%F COCV_2013__19_3_754_0
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://archive.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 | Zbl

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

[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 | Zbl

[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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

Cité par Sources :