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.

我们开发了一种基本方法，以给出当时在论文（P. Cannarsa，W。Cheng，K。Wang，J。Yan，2019 [17]）中提出的Herglotz变分原理问题中的极小值的Lipschitz估计。依赖的情况。通过使用广义du Bois-Reymond引理，我们在不同的假设集下分别推导了Erdmann条件和Euler-Lagrange方程。作为应用，我们获得了汉密尔顿-雅各比方程的柯西问题的粘度解的表示公式$d_{Ť}ü（Ť，X）+H（Ť，X，d_Xü（Ť，X），ü（Ť，X））=0$ 并研究相关的Lax-Oleinik进化。