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Adjoint methods for obstacle problems and weakly coupledsystems of PDE

Published online by Cambridge University Press:  03 June 2013

Filippo Cagnetti ,
Diogo Gomes  and
Hung Vinh Tran
Filippo Cagnetti
Affiliation:
Departamento de Matemática Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal. [email protected]; [email protected]
Diogo Gomes
Affiliation:
Departamento de Matemática Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal. [email protected]; [email protected]
Hung Vinh Tran
Affiliation:
Department of Mathematics, University of California Berkeley, CA, 94720-3840, U.S.A; [email protected]
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Abstract

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

Keywords

Adjoint methodscell problemsHamilton − Jacobi equationsobstacle problemsweakly coupled systemsweak KAM theory
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 3 , July 2013 , pp. 754 - 779
Copyright
© EDP Sciences, SMAI, 2013

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References

Barles, G. and Perthame, B., Exit time problems in optimal control and vanishing viscosity method. SIAM J. Control Optim. 26 (1988) 1133–1148. Google Scholar
Capuzzo-Dolcetta, I. and Evans, L.C., Optimal switching for ordinary differential equations. SIAM J. Control Optim. 22 (1984) 143–161. Google Scholar
Cagnetti, F., Gomes, D. and Tran, H.V., Aubry-Mather measures in the nonconvex setting. SIAM J. Math. Anal. 43 (2011) 2601–2629. Google Scholar
Camilli, F. and Loreti, P., Comparison results for a class of weakly coupled systems of eikonal equations. Hokkaido Math. J. 37 (2008) 349–362. Google Scholar
Camilli, F., Loreti, P., and Yamada, N., Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem. Commun. Pure Appl. Anal. 8 (2009) 1291–1302. Google Scholar
Engler, H. and Lenhart, S.M., Viscosity solutions for weakly coupled systems of Hamilton-Jacobi equations. Proc. London Math. Soc. 63 (1991) 212–240. Google Scholar
Evans, L.C. and Smart, C.K., Adjoint methods for the infinity Laplacian partial differential equation. Arch. Ration. Mech. Anal. 201 (2011) 87–113. Google Scholar
Evans, L.C., Adjoint and compensated compactness methods for Hamilton-Jacobi PDE. Arch. Ration. Mech. Anal. 197 (2010) 1053–1088. Google Scholar
Gomes, D.A., A stochastic analogue of Aubry-Mather theory. Nonlinearity 15 (2002) 581–603. Google Scholar
Ishii, H. and Koike, S., Viscosity solutions for monotone systems of second-order elliptic PDEs. Commun. Partial Differ. Equ. 16 (1991) 1095–1128. Google Scholar
Ishii, K. and Yamada, N., On the rate of convergence of solutions for the singular perturbations of gradient obstacle problems. Funkcial. Ekvac. 33 (1990) 551–562. Google Scholar
P.L. Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Mathematics. Pitman (Advanced Publishing Program), Boston, Mass. 69 (1982).
P.L. Lions, G. Papanicolaou and S.R.S. Varadhan, Homogenization of Hamilton-Jacobi equations, Preliminary Version, (1988).
Tran, H.V., Adjoint methods for static Hamilton-Jacobi equations. Calc. Var. Partial Differ. Equ. 41 (2011) 301–319. Google Scholar