*Published Paper*

**Inserted:** 14 nov 2012

**Last Updated:** 14 nov 2012

**Journal:** APPLIED MATHEMATICS AND OPTIMIZATION

**Volume:** 63

**Pages:** 191-216

**Year:** 2011

**Doi:** 10.1007/s00245-010-9116-7

**Abstract:**

We formulate an Hamilton-Jacobi partial differential equation $H(x, Du(x)) = 0$ on a n dimensional manifold M, with assumptions of convexity of the sets $\{p : H(x, p) \le 0\}$ subset of $T^*(x)M$, for all x. We reduce the above problem to a simpler problem; this shows that u may be built using an asymmetric distance (this is a generalization of the "distance function" in Finsler geometry); this brings forth a 'completeness' condition, and a Hopf-Rinow theorem adapted to Hamilton-Jacobi problems. The 'completeness' condition implies that u is the unique viscosity solution to the above problem.