We study the equation $u_t+ H(x, Du) = 0 in $\R^N\yimes [0,T]$ with Hamiltonian $H(x, p)$ measurable with respect to the state variable, and convex and coercive in $p$. We introduce a notion of solution based on viscosity test functions, appropriate averages in measure-theoretic sense of the Hamiltonian, and $t$-partial-sup-convolutions. We get existence results, comparison principles, and stability properties. We show that our solutions are uniform limits of viscosity solutions (in the sense of Crandall and Lions) of approximated continuous equations.
Time-dependent measurable Hamilton-Jacobi equations
CAMILLI, FABIO;
2005-01-01
Abstract
We study the equation $u_t+ H(x, Du) = 0 in $\R^N\yimes [0,T]$ with Hamiltonian $H(x, p)$ measurable with respect to the state variable, and convex and coercive in $p$. We introduce a notion of solution based on viscosity test functions, appropriate averages in measure-theoretic sense of the Hamiltonian, and $t$-partial-sup-convolutions. We get existence results, comparison principles, and stability properties. We show that our solutions are uniform limits of viscosity solutions (in the sense of Crandall and Lions) of approximated continuous equations.File in questo prodotto:
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