Advances in Differential Equations
- Adv. Differential Equations
- Volume 16, Number 7/8 (2011), 691-724.
Metric formulae for nonconvex Hamilton--Jacobi equations and applications
Abstract
We consider the Hamilton-Jacobi equation $H(x,Du)=0$ in $\mathbb R^n$, with $H$ not enjoying any convexity properties in the second variable. Our aim is to establish existence and nonexistence theorems for viscosity solutions of associated Dirichlet problems, find representation formulae and prove comparison principles. Our analysis is based on the introduction of a metric intrinsically related to the $0$--sublevels of the Hamiltonian, given by an inf-sup game theoretic formula. We also study the case where the equation is critical; i.e., $H(x,Du)= - \varepsilon$ does not admit any viscosity subsolution, for $\varepsilon >0$.
Article information
Source
Adv. Differential Equations, Volume 16, Number 7/8 (2011), 691-724.
Dates
First available in Project Euclid: 17 December 2012
Permanent link to this document
https://projecteuclid.org/euclid.ade/1355703203
Mathematical Reviews number (MathSciNet)
MR2829501
Zentralblatt MATH identifier
1235.49059
Subjects
Primary: 49L25: Viscosity solutions
Citation
Marigonda, A.; Siconolfi, A. Metric formulae for nonconvex Hamilton--Jacobi equations and applications. Adv. Differential Equations 16 (2011), no. 7/8, 691--724. https://projecteuclid.org/euclid.ade/1355703203