Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I : regularity

Mennucci, Andrea C. G.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 3, pp. 426-451

We formulate an Hamilton-Jacobi partial differential equation

$$H(x,Du(x\left)\right)=0$$

on a $n$ dimensional manifold $M$, with assumptions of convexity of $H(x,·)$ and regularity of $H$ (locally in a neighborhood of $\{H=0\}$ in $T^{*}M$); we define the “min solution” $u$, a generalized solution; to this end, we view $T^{*}M$ as a symplectic manifold. The definition of “min solution” is suited to proving regularity results about $u$; in particular, we prove in the first part that the closure of the set where $u$ is not regular may be covered by a countable number of $n-1$ dimensional manifolds, but for a ${\mathscr{H}}^{n-1}$ negligeable subset. These results can be applied to the cutlocus of a $C^2$ submanifold of a Finsler manifold.

Classification : 49L25, 53C22, 53C60
Keywords: Hamilton-Jacobi equations, conjugate points

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