Herglotz' variational principle and Lax-Oleinik evolution

WedevelopanelementarymethodtogiveaLipschitzestimateforthemin- imizers in the problem of Herglotz’ variational principle proposed in [17] in the time- dependent case. We deduce Erdmann’s condition and the Euler-Lagrange equation sep- arately 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 Dtu(t, x) + H(t, x, Dxu(t, x), u(t, x)) = 0 and study the related Lax-Oleinik evolution