Advances in Differential Equations

Metric formulae for nonconvex Hamilton--Jacobi equations and applications

A. Marigonda and A. Siconolfi

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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

Adv. Differential Equations, Volume 16, Number 7/8 (2011), 691-724.

First available in Project Euclid: 17 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 49L25: Viscosity solutions


Marigonda, A.; Siconolfi, A. Metric formulae for nonconvex Hamilton--Jacobi equations and applications. Adv. Differential Equations 16 (2011), no. 7/8, 691--724.

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