Herglotz' variational principle and Lax-Oleinik evolution
Abstract
We develop an elementary method to give a Lipschitz estimate for the minimizers in the problem of Herglotz' variational principle proposed in CCWY2018 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 align* Dtu(t,x)+H(t,x,Dxu(t,x),u(t,x))=0 align* and study the related Lax-Oleinik evolution.
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