Open Access
Translator Disclaimer
2003 Hamilton-Jacobi equations with measurable dependence on the state variable
Fabio Camilli, Antonio Siconolfi
Adv. Differential Equations 8(6): 733-768 (2003). DOI: 10.57262/ade/1355926832


We study the Hamilton-Jacobi equation \[H(x,Du)=0 , \] where $H(x,p)$ is assumed to be measurable in $x$, quasiconvex and continuous in $p$. The notion of viscosity solution is adapted to the measurable setting making use of suitable measure--theoretic devices. We obtain integral representation formulae generalizing the ones valid for continuous equations, comparison principles and uniqueness results. We examine stability properties of the new definition and present two approximation procedures: the first one is based on a regularization of the Hamiltonian by mollification and in the second one the approximating sequence is made up by minimizers of certain variational integrals.


Download Citation

Fabio Camilli. Antonio Siconolfi. "Hamilton-Jacobi equations with measurable dependence on the state variable." Adv. Differential Equations 8 (6) 733 - 768, 2003.


Published: 2003
First available in Project Euclid: 19 December 2012

zbMATH: 1036.35052
MathSciNet: MR1969652
Digital Object Identifier: 10.57262/ade/1355926832

Primary: 35F20
Secondary: 35D05 , 49L20 , 49L25

Rights: Copyright © 2003 Khayyam Publishing, Inc.


Vol.8 • No. 6 • 2003
Back to Top