Bernoulli’s Problem for the Infinity-Laplacian Near a Set with Positive Reach

We consider the exterior as well as the interior free-boundary Bernoulli problem associated with the infinity-Laplacian under a non-autonomous boundary condition. Recall that the Bernoulli problem involves two domains: one is given, the other is unknown. Concerning the exterior problem we assume tha...

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Bibliographic Details
Published in:Symmetry (Basel) 2019-04, Vol.11 (4), p.472
Main Author: Greco, Antonio
Format: Article
Language:English
Subjects:
Boundary conditions
Domains
free-boundary problems
Infinity
infinity-Laplacian
Mathematical functions
Neighborhoods
Uniqueness
Viscosity
viscosity solutions
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Summary:We consider the exterior as well as the interior free-boundary Bernoulli problem associated with the infinity-Laplacian under a non-autonomous boundary condition. Recall that the Bernoulli problem involves two domains: one is given, the other is unknown. Concerning the exterior problem we assume that the given domain has a positive reach, and prove an existence and uniqueness result together with an explicit representation of the solution. Concerning the interior problem, we obtain a similar result under the assumption that the complement of the given domain has a positive reach. In particular, for the interior problem we show that uniqueness holds in contrast to the usual problem associated to the Laplace operator.
ISSN:2073-8994
2073-8994
DOI:10.3390/sym11040472