Monge solutions for discontinuous hamiltonians

Briani, Ariela; Davini, Andrea

ESAIM: Control, Optimisation and Calculus of Variations, Tome 11 (2005) no. 2, pp. 229-251.

We consider an Hamilton-Jacobi equation of the form

$$H(x,Du)=0x\in \Omega \subset {ℝ}^N,\left(1\right)$$

where $H(x,p)$ is assumed 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 equation (1) coupled with Dirichlet boundary conditions, and a stability result. The relation among Monge and Lipschitz subsolutions is also discussed.

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