The set of continuous viscosity solutions of the infinity Laplace
equation $-\bigtriangleup^N_{\infty}w(x) = f(x)$ with generally
sign-changing right-hand-side $f$ in a bounded domain is analyzed.
The existence of a least and a greatest continuous viscosity
solutions up to the boundary is proved through a Perron's
construction by means of a strict comparison principle. These
extremal solutions are proved to be absolutely extremal solutions.

Southern California Analysis and PDE Conference Day 1

A geometric realization of an integrable system is an evolution of curves on a manifold M, invariant under the action of a group G, and such that it becomes the integrable system when the action of G is mod out. The best known geometric realization is the Vortex filament flow (VF), a flow of curves in Euclidean space which is invariant under the Euclidean group. The VF equation becomes the nonlinear Shrodinger equation when written in terms of the natural curvatures of the flow - Hasimoto proved this way the integrability of VF -. In this talk I will review what is known about the classical geometry of curves in homogeneous spaces and its relation to different types of integrable systems. In particular we will talk about how one can reduce different Hamiltonian structures to the space of curvatures (Euclidean, projective, conformal, etc) and how those reductions indicates the existence of biHamiltonian (integrable) systems. We will also describe how curvatures of Schwarzian type for curves in homogeneous spaces usually describe evolutions of KdV type. I will present background material so the talk should be accessible, at least in part, to different audiences.