Regularity of the free boundary for the obstacle problem for the fractional Laplacian with drift
Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 3, pp. 533-570.

We establish the C1+γ-Hölder regularity of the regular free boundary in the stationary obstacle problem defined by the fractional Laplace operator with drift in the subcritical regime. Our method of the proof consists in proving a new monotonicity formula and an epiperimetric inequality. Both tools generalizes the original ideas of G. Weiss in [15] for the classical obstacle problem to the framework of fractional powers of the Laplace operator with drift. Our study continues the earlier research [12], where two of us established the optimal interior regularity of solutions.

DOI : 10.1016/j.anihpc.2016.03.001
Classification : 35R35, 60G22
Mots-clés : Obstacle problem, Fractional Laplacian with drift, Free boundary regularity, Monotonicity formulas, Epiperimetric inequality, Symmetric stable process 
@article{AIHPC_2017__34_3_533_0,
     author = {Garofalo, Nicola and Petrosyan, Arshak and Pop, Camelia A. and Smit Vega Garcia, Mariana},
     title = {Regularity of the free boundary for the obstacle problem for the fractional {Laplacian} with drift},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {533--570},
     publisher = {Elsevier},
     volume = {34},
     number = {3},
     year = {2017},
     doi = {10.1016/j.anihpc.2016.03.001},
     zbl = {1365.35230},
     mrnumber = {3633735},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2016.03.001/}
}
TY  - JOUR
AU  - Garofalo, Nicola
AU  - Petrosyan, Arshak
AU  - Pop, Camelia A.
AU  - Smit Vega Garcia, Mariana
TI  - Regularity of the free boundary for the obstacle problem for the fractional Laplacian with drift
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2017
SP  - 533
EP  - 570
VL  - 34
IS  - 3
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2016.03.001/
DO  - 10.1016/j.anihpc.2016.03.001
LA  - en
ID  - AIHPC_2017__34_3_533_0
ER  - 
%0 Journal Article
%A Garofalo, Nicola
%A Petrosyan, Arshak
%A Pop, Camelia A.
%A Smit Vega Garcia, Mariana
%T Regularity of the free boundary for the obstacle problem for the fractional Laplacian with drift
%J Annales de l'I.H.P. Analyse non linéaire
%D 2017
%P 533-570
%V 34
%N 3
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2016.03.001/
%R 10.1016/j.anihpc.2016.03.001
%G en
%F AIHPC_2017__34_3_533_0
Garofalo, Nicola; Petrosyan, Arshak; Pop, Camelia A.; Smit Vega Garcia, Mariana. Regularity of the free boundary for the obstacle problem for the fractional Laplacian with drift. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 3, pp. 533-570. doi : 10.1016/j.anihpc.2016.03.001. http://www.numdam.org/articles/10.1016/j.anihpc.2016.03.001/

[1] Applebaum, D. Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, vol. 116, Cambridge University Press, Cambridge, 2009 | MR | Zbl

[2] Caffarelli, L.A.; Salsa, S.; Silvestre, L. Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., Volume 171 (2008), pp. 425–461 | DOI | MR | Zbl

[3] Caffarelli, L.A.; Silvestre, L. An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., Volume 32 (2007) no. 7–9, pp. 1245–1260 | MR | Zbl

[4] Daskalopoulos, P.; Feehan, P.M.N. Existence, uniqueness, and global regularity for variational inequalities and obstacle problems for degenerate elliptic partial differential operators in mathematical finance | arXiv

[5] Evans, L.C. Partial Differential Equations, American Mathematical Society, Providence, RI, 1998 | MR | Zbl

[6] Fabes, E.B.; Kenig, C.E.; Serapioni, R.P. The local regularity of solutions of degenerate elliptic equations, Commun. Partial Differ. Equ., Volume 7 (1982), pp. 77–116 | DOI | MR | Zbl

[7] Garofalo, N.; Petrosyan, A.; Smit Vega Garcia, M. An epiperimetric inequality approach to the regularity of the free boundary in the Signorini problem with variable coefficients | arXiv | DOI | Zbl

[8] Koch, H.; Rüland, A.; Shi, W. The variable coefficient thin obstacle problem: Carleman inequalities | arXiv | DOI | Zbl

[9] Koch, H.; Rüland, A.; Shi, W. The variable coefficient thin obstacle problem: optimal regularity and regularity of the regular free boundary | arXiv | Numdam | Zbl

[10] Molchanov, S.A.; Ostrovskii, E. Symmetric stable processes as traces of degenerate diffusion processes, Theory Probab. Appl., Volume 14 (1969) no. 1, pp. 128–131 | DOI | MR | Zbl

[11] Nekvinda, A. Characterization of traces of the weighted Sobolev space W1,p(Ω,dMϵ) on M , Czechoslov. Math. J., Volume 43 (1993) no. 4, pp. 695–711 | MR | Zbl

[12] Petrosyan, A.; Pop, C.A. Optimal regularity of solutions to the obstacle problem for the fractional Laplacian with drift, J. Funct. Anal., Volume 268 (2015) no. 2, pp. 417–472 | DOI | MR | Zbl

[13] Silvestre, L. Regularity of the obstacle problem for a fractional power of the Laplace operator, Commun. Pure Appl. Math., Volume 60 (2007) no. 1, pp. 67–112 | DOI | MR | Zbl

[14] Weiss, G.S. Partial regularity for weak solutions of an elliptic free boundary problem, Commun. Partial Differ. Equ., Volume 23 (1998) no. 3–4, pp. 439–455 | MR | Zbl

[15] Weiss, G.S. A homogeneity improvement approach to the obstacle problem, Invent. Math., Volume 138 (1999), pp. 23–50 | DOI | MR | Zbl

Cité par Sources :