no. 3
Regularity in a one-phase free boundary problem for the fractional Laplacian
Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 3, pp. 335-367.

For a one-phase free boundary problem involving a fractional Laplacian, we prove that “flat free boundaries” are C 1,α . We recover the regularity results of Caffarelli for viscosity solutions of the classical Bernoulli-type free boundary problem with the standard Laplacian.

@article{AIHPC_2012__29_3_335_0,
     author = {De Silva, D. and Roquejoffre, J.M.},
     title = {Regularity in a one-phase free boundary problem for the fractional {Laplacian}},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {335--367},
     publisher = {Elsevier},
     volume = {29},
     number = {3},
     year = {2012},
     doi = {10.1016/j.anihpc.2011.11.003},
     zbl = {1251.35178},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2011.11.003/}
}
TY  - JOUR
AU  - De Silva, D.
AU  - Roquejoffre, J.M.
TI  - Regularity in a one-phase free boundary problem for the fractional Laplacian
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2012
SP  - 335
EP  - 367
VL  - 29
IS  - 3
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2011.11.003/
DO  - 10.1016/j.anihpc.2011.11.003
LA  - en
ID  - AIHPC_2012__29_3_335_0
ER  - 
%0 Journal Article
%A De Silva, D.
%A Roquejoffre, J.M.
%T Regularity in a one-phase free boundary problem for the fractional Laplacian
%J Annales de l'I.H.P. Analyse non linéaire
%D 2012
%P 335-367
%V 29
%N 3
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2011.11.003/
%R 10.1016/j.anihpc.2011.11.003
%G en
%F AIHPC_2012__29_3_335_0
De Silva, D.; Roquejoffre, J.M. Regularity in a one-phase free boundary problem for the fractional Laplacian. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 3, pp. 335-367. doi : 10.1016/j.anihpc.2011.11.003. http://archive.numdam.org/articles/10.1016/j.anihpc.2011.11.003/

[1] H.W. Alt, L.A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105-144 | EuDML | Zbl

[2] L.A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part I: Lipschitz free boundaries are C 1,α , Rev. Mat. Iberoam. 3 no. 2 (1987), 139-162 | EuDML | Zbl

[3] L.A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part II: Flat free boundaries are Lipschitz, Comm. Pure Appl. Math. 42 no. 1 (1989), 55-78 | Zbl

[4] L.A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part III: Existence theory, compactness, and dependence on X, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 15 no. 4 (1988), 583-602 | EuDML | Numdam | Zbl

[5] L.A. Caffarelli, X. Cabre, Fully Nonlinear Elliptic Equations, Colloquium Publications vol. 43, Amer. Math. Soc., Providence, RI (1995) | Zbl

[6] L.A. Caffarelli, J.-M. Roquejoffre, O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 (2010), 1111-1144 | Zbl

[7] L.A. Caffarelli, J.-M. Roquejoffre, Y. Sire, Variational problems with free boundaries for the fractional Laplacian, J. Eur. Math. Soc. (JEMS) 12 (2010), 1151-1179 | EuDML | Zbl

[8] L.A. Caffarelli, S. Salsa, A Geometric Approach to Free Boundary Problems, Grad. Stud. Math. vol. 68, Amer. Math. Soc., Providence, RI (2005) | Zbl

[9] L.A. Caffarelli, S. Salsa, L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math. 171 (2008), 425-461 | Zbl

[10] L.A. Caffarelli, L. Silvestre, An extension problem for the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), 1245-1260 | Zbl

[11] D. De Silva, D. Jerison, A singular energy minimising free boundary, J. Reine Angew. Math. 635 (2009), 1-21 | Zbl

[12] D. De Silva, Free boundary regularity for a problem with right hand side, Interfaces Free Bound. 13 (2011), 223-238 | Zbl

[13] O. Savin, Small perturbation solutions for elliptic equations, Comm. Partial Differential Equations 32 (2007), 557-578 | Zbl

[14] L. Wang, Compactness methods for certain degenerate elliptic equations, J. Differential Equations 107 (1994), 341-350 | Zbl

Cité par Sources :