Calculus of Variations and Geometric Measure Theory

A. C. G. Mennucci

Regularity and Variationality of Solutions to Hamilton-Jacobi Equations. Part II: Variationality, Existence, Uniqueness.

created by mennucci on 14 Nov 2012


Published Paper

Inserted: 14 nov 2012
Last Updated: 14 nov 2012

Volume: 63
Pages: 191-216
Year: 2011
Doi: 10.1007/s00245-010-9116-7


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.