Convexity and Generalized Convexity Methods for the Study of Hamilton-Jacobi Equations

We present a digest of recent results about the first-order HamiltonJacobi equation. We use explicit formulas of the Hopf and Lax-Oleinik types, stressing the role of quasiconvex duality: here the usual Fenchel conjugacy is replaced with quasiconvex conjugacies known from some years and the usual inf-convolution is replaced by the sublevel convolution. The role of the full theory of variational convergences (epi-convergence and sublevel convergence) is put in light for the verification of initial conditions. We observe that duality methods and variational convergences are not limited to the case of finite-valued functions as in the classical approaches to Hamilton-Jacobi equations. This extension allows to deal with problems arising from various cases, such as optimal control problems, attainability or obstacle problems.

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