We consider the Hamilton-Jacobi equation $H(x,Du)=0$ in $\mathbb R^n$, with $H$ non 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 $\varepsilon >0$.

Metric formulae for nonconvex Hamilton-Jacobi equations and applications

MARIGONDA, ANTONIO;
2011-01-01

Abstract

We consider the Hamilton-Jacobi equation $H(x,Du)=0$ in $\mathbb R^n$, with $H$ non 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 $\varepsilon >0$.
2011
Nonconvex Hamilton-Jacobi equations; viscosity solutions; Aubry-Mather theory
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11562/344804
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