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

Barles, Guy; Perthame, Benoît (1990), Comparison principle for dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations, Applied Mathematics and Optimization, 21, 1, p. 21-44. http://dx.doi.org/10.1007/BF01445155

Type
Article accepté pour publication ou publié
Date
1990
Journal name
Applied Mathematics and Optimization
Volume
21
Number
1
Publisher
Springer
Pages
21-44
Publication identifier
Metadata
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Author(s)
Barles, Guy
Perthame, Benoît cc
Abstract (EN)
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)=0inΩ,Max(H(x,u,Du);u−φ)⩾0on∂Ω,Min(H(x,u,Du);u−φ)⩽0on∂Ω. 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.
Subjects / Keywords
PDE; Hamilton-Jacobi equations