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arXiv · 1907.05769

Herglotz' variational principle and Lax-Oleinik evolution

Piermarco Cannarsa·Wei Cheng·Liang Jin·Kaizhi Wang·Jun Yan
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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.

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Piermarco Cannarsa, Wei Cheng, Liang Jin, Kaizhi Wang, Jun Yan. 2019-07-12. Herglotz' variational principle and Lax-Oleinik evolution. https://arxiv.org/abs/1907.05769

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