We consider the following evolutionary Hamilton-Jacobi equation with initial condition: { ∂tu(x, t) +H(x, u(x, t), ∂xu(x, t)) = 0, u(x, 0) = φ(x). Under some assumptions on H(x, u, p) with respect to p and u, we provide a variational principle on the evolutionary Hamilton-Jacobi equation. By introducing an implicitly defined solution semigroup, we extend Fathi’s weak KAM theory to certain more ...

The geometric structure of the Hamilton-Jacobi equation is analyzed by using symplectic geometry. Generating function of symplectic transform plays an important role. It will be shown that the Hamilton-Jacobi equation possesses important geometric properties such as existence condition and maximality of the stabilizing solution, which are well-known in the Riccati equation.

Stationary quasi-P eikonal equations, stationary Hamilton-Jacobi equations, arise from the asymptotic approximation of anisotropic wave propagation. A paraxial formulation of the quasi-P eikonal equation results in a paraxial quasi-P eikonal equation, an evolution Hamilton-Jacobi equation in a preferred direction, which provides a fast and efficient way for computing viscosity solutions of quas...

We establish well-posedness of a class of first order Hamilton-Jacobi equation in geodesic metric spaces. The result is then applied to solve a Hamilton-Jacobi equation in the Wasserstein space of probability measures, which arises from the variational formulation of a compressible Euler equation.

Classic, and Explicit Turnpikes Paolo Guasoni · Constantinos Kardaras · Scott Robertson · Hao Xing Received: date / Accepted: date Abstract Portfolio turnpikes state that, as the investment horizon increases, optimal portfolios for generic utilities converge to those of isoelastic utilities. This paper proves three kinds of turnpikes. In a general semimartingale setting, the abstract turnpike s...

We discuss a class of time-dependent Hamilton-Jacobi equations, where an unknown function of time is intended to keep the maximum of the solution to the constant value 0. Our main result is that the full problem has a unique viscosity solution, which is in fact classical. The motivation is a selection-mutation model which, in the limit of small diffusion, exhibits concentration on the zero leve...

In this work, we propose a high resolution Alternating Evolution Discontinuous Galerkin (AEDG) method to solve Hamilton-Jacobi equations. The construction of the AEDG method is based on an alternating evolution system of the Hamilton-Jacobi equation, following the previous work [H. Liu, M. Pollack and H. Saran, SIAM J. Sci. Comput. 35(1), (2013) 122–149] on AE schemes for Hamilton-Jacobi equati...