Herglotz' variational principle and Lax-Oleinik evolution

We develop an elementary method to give a Lipschitz estimate for the minimizers in the problem of Herglotz' variational principle proposed in the paper (P. Cannarsa, W. Cheng, K. Wang, J. Yan, 2019 [17]) in the time-dependent case. We deduce Erdmann's condition and the Euler-Lagrange equation separately under different sets of assumptions, by using a generalized du Bois-Reymond lemma. As an application, we obtain a representation formula for the viscosity solution of the Cauchy problem for the Hamilton-Jacobi equation D(t)u(t, x) + H(t, x, D(x)u(t, x), u(t, x)) = 0 and study the related Lax-Oleinik evolution. (C) 2020 Elsevier Masson SAS. All rights reserved.