Bernoulli Variational Problem and Beyond
- 271 Downloads
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.
Unable to display preview. Download preview PDF.
- 3.Berestycki, H., Caffarelli, L. A., Nirenberg, L.: Uniform estimates for regularization of free boundary problems, in Analysis and partial differential equations, vol. 122 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 567–619, 1990Google Scholar
- 4.Braides, A.: Γ-convergence for beginners, vol. 22 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford 2002Google Scholar
- 5.Caffarelli, L.A.: A Harnack inequality approach to the regularity of free boundaries. I. Lipschitz free boundaries are C 1, α, Rev. Mat. Iberoamericana, vol. 3, pp. 139–162 1987Google Scholar
- 6.Caffarelli, L.A.: A Harnack inequality approach to the regularity of free boundaries. III. Existence theory, compactness, and dependence on X. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15(1988), 583–602 (1989)Google Scholar
- 19.www.freefem.org. Accessed 23 Feb 2013