Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications by Yves Achdou Guy Barles Hitoshi Ishii & Grigory L. Litvinov
Author:Yves Achdou, Guy Barles, Hitoshi Ishii & Grigory L. Litvinov
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg
Now, since
v λ satisfies in the viscosity sense
which implies, together with the coercivity of H, the equi-Lipschitz continuity of . Thus the collections and of functions on are relatively compact in . We may select a sequence such that
for some functions as j → ∞, where the convergences to v and w are uniform on . Observe that for all ,
which shows that w is constant on . If we write this constant as a, then we see by Proposition 1.9 that v is a viscosity solution of (EVP). This completes the proof of (i). □
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