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    Bernoulli Variational Problem and Beyond

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    Type
    Article
    Authors
    Lorz, Alexander
    Markowich, Peter A. cc
    Perthame, Benoît
    KAUST Department
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Applied Mathematics and Computational Science Program
    Date
    2013-12-17
    Online Publication Date
    2013-12-17
    Print Publication Date
    2014-05
    Permanent link to this record
    http://hdl.handle.net/10754/563150
    
    Metadata
    Show full item record
    Abstract
    The question of 'cutting the tail' of the solution of an elliptic equation arises naturally in several contexts and leads to a singular perturbation problem under the form of a strong cut-off. We consider both the PDE with a drift and the symmetric case where a variational problem can be stated. It is known that, in both cases, the same critical scale arises for the size of the singular perturbation. More interesting is that in both cases another critical parameter (of order one) arises that decides when the limiting behaviour is non-degenerate. We study both theoretically and numerically the values of this critical parameter and, in the symmetric case, ask if the variational solution leads to the same value as for the maximal solution of the PDE. Finally we propose a weak formulation of the limiting Bernoulli problem which incorporates both Dirichlet and Neumann boundary condition. © 2013 Springer-Verlag Berlin Heidelberg.
    Citation
    Lorz, A., Markowich, P., & Perthame, B. (2013). Bernoulli Variational Problem and Beyond. Archive for Rational Mechanics and Analysis, 212(2), 415–443. doi:10.1007/s00205-013-0707-8
    Sponsors
    The authors wish to thank the Fondation Sciences Mathematiques de Paris for the support of AL and PM. The authors also thank Frederic Hecht for his decisive advice on the numerics based on FreeFEM++.
    Publisher
    Springer Nature
    Journal
    Archive for Rational Mechanics and Analysis
    DOI
    10.1007/s00205-013-0707-8
    ae974a485f413a2113503eed53cd6c53
    10.1007/s00205-013-0707-8
    Scopus Count
    Collections
    Articles; Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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