Graph selectors and viscosity solutions on lagrangian manifolds
ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 4, p. 795-815

Let Λ be a lagrangian submanifold of T * X for some closed manifold X. Let S(x,ξ) be a generating function for Λ which is quadratic at infinity, and let W(x) be the corresponding graph selector for Λ, in the sense of Chaperon-Sikorav-Viterbo, so that there exists a subset X 0 X of measure zero such that W is Lipschitz continuous on X, smooth on XX 0 and (x,W/x(x))Λ for XX 0 . Let H(x,p)=0 for (x,p)Λ. Then W is a classical solution to H(x,W/x(x))=0 on XX 0 and extends to a Lipschitz function on the whole of X. Viterbo refers to W as a variational solution. We prove that W is also a viscosity solution under some simple and natural conditions. We also prove that these conditions are satisfied in many cases, including certain commonly occuring cases where H(x,p) is not convex in p.

DOI : https://doi.org/10.1051/cocv:2006023
Classification:  49L25,  53D12
Keywords: viscosity solution, lagrangian manifold, graph selector
@article{COCV_2006__12_4_795_0,
     author = {McCaffrey, David},
     title = {Graph selectors and viscosity solutions on lagrangian manifolds},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {12},
     number = {4},
     year = {2006},
     pages = {795-815},
     doi = {10.1051/cocv:2006023},
     zbl = {1114.49030},
     mrnumber = {2266819},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2006__12_4_795_0}
}
McCaffrey, David. Graph selectors and viscosity solutions on lagrangian manifolds. ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 4, pp. 795-815. doi : 10.1051/cocv:2006023. http://www.numdam.org/item/COCV_2006__12_4_795_0/

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