Generalizing Hopf and Lax-Oleînik formulae via conjugate integral

Using the second Fenchel conjugate transform the conjugate integral sums and the conjugate integral are introduced. An estimate of speed of convergence of the sums to the integral is obtained. In the case of a convex integrant the conjugate integral reduces to the Riemannian one. It is proved that the Fenchel conjugate transform of the conjugate integral with variable upper limit provides a formula for the viscosity solution to a Hamilton-Jacobi equation in which the Hamiltonian depends both on time and the gradient of the unknown function. In the autonomous case the obtained formula coincides with Hopf's one. Two examples are considered in which an application of the conjugate integral allows to find viscosity solutions explicitly. It is shown how the extension of the Lax-Oleînik formula to the nonautonomous case may be obtained using the generalized Hopf formula.