Abstract
The classical Hopf-Lax formula for an explicit solution of a Hamilton-Jacobi equation is extended to equations of the formu t +H(u,Du)=0 with terminal datau(T, x)=g(x) assumed to be merely quasiconvex, i.e., having convex level sets. Using a new quasiconvex conjugateg *γ,p), the formula is given byu(t,x)=(g *(γ,p)-(T−t)H(γp))*. We give a direct proof thatu is a Subbotin minimax solution of the problem. The first-order obstacle problem associated with optimal control inL ∞ is also studied and an explicit solution given.
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Research supported in part by grants DMS-9300805 and DMS-9532030 from the National Science Foundation.
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Barron, E.N., Jensen, R. & Liu, W. Explicit solution of some first-order PDE's. Journal of Dynamical and Control Systems 3, 149–164 (1997). https://doi.org/10.1007/BF02465892
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DOI: https://doi.org/10.1007/BF02465892