Herglotz' variational principle and Lax-Oleinik evolution

2019/07/12 by Cannarsa, Piermarco, Cheng, Wei, Jin, Liang +2 · 2 citations
#Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1907.05769

Abstract

We develop an elementary method to give a Lipschitz estimate for the minimizers in the problem of Herglotz' variational principle proposed in \citeCCWY2018 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_tu(t,x)+H(t,x,D_xu(t,x),u(t,x))=0 and study the related Lax-Oleinik evolution.

Cited by

• Geometric Formulation of Combined Conservative Dissipative Mechanics via Contact Hamiltonian Dynamics Symmetries, Reduction, and Variational Integrators
• Quantitative homogenization of convex Hamilton-Jacobi equations with u/ε-periodic Hamiltonians

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