Geodesic
Variational inequalities for the spectral fractional Laplacian
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 3 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper we study obstacle problems for the Navier (spectral) fractional Laplacian $(-\Delta_{\Omega})^s$ of order $s \in (0,1)$ in a bounded domain $\Omega\subset \mathrm{R}^n$. 
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R. Musina; A. I. Nazarov. Variational inequalities for the spectral fractional Laplacian. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 3. http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_3_a1/

[1] R. Alvarado, D. Brigham, V. Maz'ya, M. Mitrea, E. Ziade, “On the regularity of domains satisfying a uniform hour-glass condition and a sharp version of the Hopf–Oleinik boundary point principle”, J. Math. Sci., 176 (2011), 281–360 | DOI | MR | Zbl

[2] B. Barrios, A. Figalli, X. Ros-Oton, Global regularity for the free boundary in the obstacle problem for the fractional Laplacian, 2015, arXiv: 1506.04684

[3] G. Buttazzo, G. Dal Maso, “An existence result for a class of shape optimization problems”, Arch. Rational Mech. Anal., 122:2 (1993), 183–195 | DOI | MR | Zbl

[4] L. Caffarelli, A. Figalli, “Regularity of solutions to the parabolic fractional obstacle problem”, J. Reine Angew. Math., 680 (2013), 191–233 | MR | Zbl

[5] L. Caffarelli, L. Silvestre, “An extension problem related to the fractional Laplacian”, Commun. Part. Differ. Equations, 32:7–9 (2007), 1245–1260 | DOI | MR | Zbl

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

[7] A. Capella, J. Davila, L. Dupaigne, Y. Sire, “Regularity of radial extremal solutions for some non-local semilinear equations”, Commun. Part. Differ. Equations, 36:8 (2011), 1353–1384 | DOI | MR | Zbl

[8] G. Dal Maso, An Introduction to $\Gamma$-Convergence, Progress in Nonlinear Differential Equations and Their Applications, 8, Birkhauser, Boston, MA, 1993 | MR | Zbl

[9] N. Dunford, J. T. Schwartz, Linear Operators, v. II, Wiley, New York, 1988 | MR

[10] N. Garofalo, A. Petrosyan, “Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem”, Invent. Math., 177:2 (2009), 415–461 | DOI | MR | Zbl

[11] G. Grubb, “Regularity of spectral fractional Dirichlet and Neumann problems”, Math. Nachr., 289:7 (2015), 831–844 | DOI | MR

[12] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Birkhauser, Basel, 2006 | MR | Zbl

[13] A. Iannizzotto, S. Mosconi, M. Squassina, “Hs versus CO-weighted minimizers”, Nonlinear Differ. Equations Appl., 22:3 (2015), 477–497 | DOI | MR | Zbl

[14] L. I. Kamynin, B. N. Himcenko, “Theorems of Giraud type for second order equations with a weakly degenerate nonnegative characteristic part”, Sib. Math. J., 18 (1977), 76–91 | DOI | MR | Zbl

[15] D. Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, reprint of the 1980 original, Classics in Applied Mathematics, 31, SIAM, Philadelphia, PA, 2000 | MR | Zbl

[16] H. Lewy, G. Stampacchia, “On the regularity of the solution of a variational inequality”, Commun. Pure Appl. Math., 22 (1969), 153–188 | DOI | MR | Zbl

[17] H. Lewy, G. Stampacchia, “On the smoothness of superharmonics which solve a minimum problem”, J. Anal. Math., 23 (1970), 227–236 | DOI | MR | Zbl

[18] R. Musina, A. I. Nazarov, “On fractional Laplacians”, Commun. Part. Differ. Equations, 39:9 (2014), 1780–1790 | DOI | MR | Zbl

[19] R. Musina, A. I. Nazarov, “On fractional Laplacians 2”, Ann. Inst. H. Poincaré Anal. Non Lineairé, 33:6 (2016), 1667–1673 | DOI | MR | Zbl

[20] R. Musina, A. I. Nazarov, K. Sreenadh, “Variational inequalities for the fractional Laplacian”, Potential Anal., 2016 | DOI | MR

[21] A. Petrosyan, C. A. Pop, “Optimal regularity of solutions to the obstacle problem for the fractional Laplacian with drift”, J. Funct. Anal., 268:2 (2015), 417–472 | DOI | MR | Zbl

[22] T. Runst, W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, de Gruyter Series in Nonlinear Analysis and Applications, 3, de Gruyter, Berlin, 1996 | MR | Zbl

[23] R. Servadei, E. Valdinoci, “Lewy–Stampacchia type estimates for variational inequalities driven by (non) local operators”, Rev. Mat. Iberoam., 29:3 (2013), 1091–1126 | DOI | MR | Zbl

[24] L. Silvestre, “Regularity of the obstacle problem for a fractional power of the Laplace operator”, Commun. Pure Appl. Math., 60:1 (2007), 67–112 | DOI | MR | Zbl

[25] P. R. Stinga, J. L. Torrea, “Extension problem and Harnack's inequality for some fractional operators”, Commun. Part. Differ. Equations, 35:11 (2010), 2092–2122 | DOI | MR | Zbl

[26] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Deutscher Verlag der Wissenschaften, Berlin, 1978 | MR | Zbl