# Metric Viscosity Solutions of Hamilton-Jacobi Equations Depending on Local Slopes.

### (Georgia Tech.)

**Event:** ERC Workshop on Optimal Transportation and Applications

**Date:** Oct 29, 2014,
**time:** 09:55

**Abstract.** We present a theory of metric viscosity solutions which encompasses a large class of Hamiltonians. We consider time dependent problems

\[\label{a111}

\partial_tu+H(t,x,u,\vert\nabla u\vert)=0,\quad\mbox{in}\,\,(0,T)\times \Omega,

\]

\begin{equation}\label{a111BC}

\left\{

\begin{array}{ll}

&

u(t,x)=f(t,x)\quad\mbox{on}\,\,(0,T)\times \partial\Omega,

\\

&

u(0,x)=g(x)\quad\mbox{on}\,\,\Omega,

\end{array}

\right.

\end{equation}

and stationary equations. We prove a range of comparison and existence results that apply to a wide range of equations and we present a sample of

techniques that can apply in other cases. (This talk is based on a joint work with A. Swiech).