A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations

Hitoshi Ishii

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In this paper we prove a comparison result between semicontinuous viscosity subsolutions and supersolutions to Hamilton-Jacobi equations of the form $u_{t} + H (x, D u) = 0$ in $I R^{n} \times (0, T)$ where the Hamiltonian may be noncoercive in the gradient As a consequence of the comparison result and the Perron's method we get the existence of a continuous solution of this equation.