A representation formula of the viscosity solution of the contact Hamilton-Jacobi equation and its applications
Published in Chinese Annals of Mathematics, Series B, 2023
Recommended citation: P. Ni, L. Wang, and J. Yan, A representation formula of the viscosity solution of the contact Hamilton-Jacobi equation and its applications, Chinese Annals of Mathematics, Series B, to appear. https://arxiv.org/abs/2101.00446
In this work, we generalized the dynamical assumptions in
[1] K. Wang, L. Wang, and J. Yan, Variational principle for contact Hamiltonian systems and its applications, J. Math. Pures Appl., 123 (2019), 167-200.
to the standard PDE assumptions, that is, H(x,u,p) is continuous, convex and coercive in p, uniformly Lipschitz in u. As its applications, we obtain a necessary and sufficient condition for the existence of the viscosity solutions of the stationary equations. Moreover, we prove a new comparison theorem depending on the neighborhood of the projected Aubry set essentially, which is different from the one for the Hamilton-Jacobi equation independent of u. An example is constructed to show that the requirement of the neighborhood is necessary for a special class of Hamilton-Jacobi equations that do not satisfy the “proper” condition.