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We investigate the vanishing viscosity limit for Hamilton-Jacobi PDE with non-convex Hamiltonians, and present a new method to augment the standard viscosity solution approach. The main idea is to introduce a solution σ ε of the adjoint of the formal linearization, and then to integrate by parts with respect to the density σ ε. This procedure leads to a… (More)
This paper, building upon ideas of Mather, Moser, Fathi, E and others, applies PDE (partial differential equation) methods to understand the structure of certain Hamiltonian flows. The main point is that the " cell " or " corrector " PDE, introduced and solved in a weak sense by Lions, Papanicolaou and Varadhan in their study of periodic homogenization for… (More)
We propose a new method for showing C 1,α regularity for solutions of the infinity Laplacian equation and provide full details of the proof in two dimensions. The proof for dimensions n ≥ 3 depends upon some conjectured local gradient estimates for solutions of certain transformed PDE.
Our intention in this paper is to publicize and extend somewhat important work of Plotnikov [P] on the asymptotic limits of solutions of viscous regularizations of an nonlinear diffusion PDE with a cubic nonlinearity. Since the formal limit PDE is in general ill–posed, we expect that the limit solves instead a corresponding diffusion equation with… (More)
To study fine properties of certain smooth approximations u ε to a viscosity solution u of the infinity Laplacian PDE, we introduce Green's function σ ε for the linearization. We can then integrate by parts with respect to σ ε and derive various useful integral estimates. We are in particular able to use these estimates (i) to prove the everywhere… (More)
In this paper, we show the existence of a unique, regular solution to the flow of the H-system with Dirichlet boundary condition. The solution exists at least up until the time of energy concentration. If this solution satisfies a certain energy inequality, then it can be continued to a global solution with the exception of at most finitely many… (More)