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

## Wilfrid Gangbo

### (Georgia Tech.)

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
$\label{a111} \partial_tu+H(t,x,u,\vert\nabla u\vert)=0,\quad\mbox{in}\,\,(0,T)\times \Omega,$
\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.

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).

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