Regularity in a one-phase free boundary problem for the fractional Laplacian
De Silva, D. ; Roquejoffre, J.M.
Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012), p. 335-367 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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},
     volume = {29},
     year = {2012},
     pages = {335-367},
     doi = {10.1016/j.anihpc.2011.11.003},
     zbl = {1251.35178},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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) pp. 335-367. doi : 10.1016/j.anihpc.2011.11.003. http://gdmltest.u-ga.fr/item/AIHPC_2012__29_3_335_0/

[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 | Zbl 0449.35105

[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 | Zbl 0676.35085

[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 0676.35086

[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 | Numdam | Zbl 0702.35249

[5] L.A. Caffarelli, X. Cabre, Fully Nonlinear Elliptic Equations, Colloquium Publications vol. 43, Amer. Math. Soc., Providence, RI (1995) | Zbl 0834.35002

[6] L.A. Caffarelli, J.-M. Roquejoffre, O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 (2010), 1111-1144 | Zbl 1248.53009

[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 | Zbl 1221.35453

[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 1083.35001

[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 1148.35097

[10] L.A. Caffarelli, L. Silvestre, An extension problem for the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), 1245-1260 | Zbl 1143.26002

[11] D. De Silva, D. Jerison, A singular energy minimising free boundary, J. Reine Angew. Math. 635 (2009), 1-21 | Zbl 1185.35050

[12] D. De Silva, Free boundary regularity for a problem with right hand side, Interfaces Free Bound. 13 (2011), 223-238 | Zbl 1219.35372

[13] O. Savin, Small perturbation solutions for elliptic equations, Comm. Partial Differential Equations 32 (2007), 557-578 | Zbl 1221.35154

[14] L. Wang, Compactness methods for certain degenerate elliptic equations, J. Differential Equations 107 (1994), 341-350 | Zbl 0792.35067