Bernoulli Variational Problem and Beyond

Lorz, Alexander; Markowich, Peter A.; Perthame, Benoît

Bernoulli Variational Problem and Beyond

Type

Article

Authors

Lorz, Alexander
Markowich, Peter A.

Perthame, Benoît

KAUST Department

Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Applied Mathematics and Computational Science Program

Online Publication Date

2013-12-17

Print Publication Date

2014-05

Date

2013-12-17

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

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++.