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Comparison principle for Dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations. (English) Zbl 0691.49028
Summary: Under a nondegeneracy condition on the boundary, we prove a comparison principle for discontinuous viscosity sub- and supersolutions of the generalized Dirichlet boundary-value problem for a first-order Hamilton- Jacobi equation, $H(x,u,Du)=0\quad in\quad \Omega,$ $Max(H(x,u(Du);\quad u-\phi)\geq 0\quad on\quad \partial \Omega,$ $Min(H(x,u,Du);\quad u-\phi)\leq 0\quad on\quad \partial \Omega.$ For optimal control problems, we interpret this nondegeneracy as a condition on the controlled vector fields. Finally, we use this to extend classical singular perturbation results to degenerated elliptic equations.

##### MSC:
 49L99 Hamilton-Jacobi theories 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35B25 Singular perturbations in context of PDEs
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