Calculus of Variations and Geometric Measure Theory

A. Briani - A. Davini

Monge solutions for discontinuous Hamiltonians

created on 18 Sep 2003
modified by davini on 25 Oct 2005


Published Paper

Inserted: 18 sep 2003
Last Updated: 25 oct 2005

Journal: ESAIM: Control Opt. Calc. Var.
Volume: 11
Pages: 229-251
Year: 2005


We consider an Hamilton-Jacobi equation of the form $H(x,Du)=0$ in $A$, where $A$ is a subset of $R^N$ and $H(x,p)$ is Borel measurable and quasi-convex in $p$. The notion of Monge solution, introduced by Newcomb and Su, is adapted to this setting making use of suitable metric devices. We establish the comparison principle for Monge sub and supersolution, existence and uniqueness for the above equation coupled with Dirichlet boundary conditions, and a stability result. The relation among Monge and Lipschitz subsolutions is also discussed.