Calculus of Variations and Geometric Measure Theory

A. C. G. Mennucci

Regularity and Variationality of Solutions to Hamilton-Jacobi Equations. part I: regularity

created on 17 Dec 2002
modified by mennucci on 07 Jun 2013


Published Paper

Inserted: 17 dec 2002
Last Updated: 7 jun 2013

Year: 2003
Doi: 10.1051/cocv:2004014

original version DOI 10.1051cocv:2004014 ; then errata DOI 10.1051cocv:2007019 ; the 2007 version here includes the errata


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 $H(x,\cdot)$ 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 ${\cal H}^{n-1}$ negligeable subset.

These results can be applied to the cutlocus of a $C^2$ submanifold of a Finsler manifold.

Keywords: rectifiable, cutlocus