Bernoulli Variational Problem and Beyond

Alexander Lorz, Peter A. Markowich, Benoît Perthame

Research output: Contribution to journalArticlepeer-review


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.
Original languageEnglish (US)
Pages (from-to)415-443
Number of pages29
JournalArchive for Rational Mechanics and Analysis
Issue number2
StatePublished - Dec 17 2013

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: 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++.

ASJC Scopus subject areas

  • Mechanical Engineering
  • Analysis
  • Mathematics (miscellaneous)


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