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Comparison principle for dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations

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Abstract

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

$$\left\{ {\begin{array}{*{20}c} {H(x,u,Du) = 0 in \Omega ,} \\ {Max(H(x,u,Du);u - \varphi ) \geqslant 0 on \partial \Omega ,} \\ {Min(H(x,u,Du);u - \varphi ) \leqslant 0 on \partial \Omega .} \\ \end{array} } \right.$$

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.

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Authors and Affiliations

  1. Ceremade, Université de Paris IX Dauphine, Place de Lattre de Tassigny, 75775, Paris Cedex 16, France

    G. Barles

  2. CMA, Ecole Normale Supérieure, 45 Rue d'Ulm, 75230, Paris Cedex 05, France

    B. Perthame

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  1. G. Barles
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  2. B. Perthame
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Communicated by A. Bensoussan

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Barles, G., Perthame, B. Comparison principle for dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations. Appl Math Optim 21, 21–44 (1990). https://doi.org/10.1007/BF01445155

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