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 19 Aug 2005
modified on 24 Sep 2010


Accepted Paper

Inserted: 19 aug 2005
Last Updated: 24 sep 2010

Journal: Applied Mathematics and Optimization
Year: 2010

DOI: 10.1007s00245-010-9116-7


We formulate an Hamilton--Jacobi partial differential equation $$H( x, D u(x))=0$$ on a $n$ dimensional manifold $M$, with assumptions of convexity of the sets $\{p:H(x,p) < 0\}$ in $T^*_x M$, for all $x$.

In this paper 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.

When $H$ is moreover of class $C^{1,1}$, we show how the completeness condition is equivalent to a condition expressed using the characteristics equations.

Keywords: viscosity solution, Hamilton-Jacobi equation, differentiable manifold, Kuratowski convergence, asymmetric metric space, Finsler metric, Hopf-Rinow theorem, backward completeness, uniqueness of solution