We consider the Hamilton-Jacobi equation H(x, Du) = 0 in 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) = -epsilon does not admit any viscosity subsolution, for epsilon > 0.
METRIC FORMULAE FOR NONCONVEX HAMILTON-JACOBI EQUATIONS AND APPLICATIONS / A., Marigonda; Siconolfi, Antonio. - In: ADVANCES IN DIFFERENTIAL EQUATIONS. - ISSN 1079-9389. - STAMPA. - 16:7-8(2011), pp. 691-724.
METRIC FORMULAE FOR NONCONVEX HAMILTON-JACOBI EQUATIONS AND APPLICATIONS
SICONOLFI, Antonio
2011
Abstract
We consider the Hamilton-Jacobi equation H(x, Du) = 0 in 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) = -epsilon does not admit any viscosity subsolution, for epsilon > 0.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
