Herglotz' variational principle and Lax-Oleinik evolution
Analysis of PDEs
2019-07-15 v1 Dynamical Systems
Abstract
We develop an elementary method to give a Lipschitz estimate for the minimizers in the problem of Herglotz' variational principle proposed in \cite{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 \begin{align*} D_tu(t,x)+H(t,x,D_xu(t,x),u(t,x))=0 \end{align*} and study the related Lax-Oleinik evolution.
Cite
@article{arxiv.1907.05769,
title = {Herglotz' variational principle and Lax-Oleinik evolution},
author = {Piermarco Cannarsa and Wei Cheng and Liang Jin and Kaizhi Wang and Jun Yan},
journal= {arXiv preprint arXiv:1907.05769},
year = {2019}
}