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We reconsider in this paper boundary value problems for the so-called " infinity Lapla-cian " PDE and the relationships with optimal Lipschitz extensions of the boundary data. We provide some fairly elegant new proofs, which clarify and simplify previous work, and in particular draw attention to the fact that solutions may be characterized by a comparison… (More)
We demonstrate that a solution to the classical Monge–Kantorovich problem of optimally rearranging the measure µ +
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)