Geodesic
Almost minimizers for certain fractional variational problems
Algebra i analiz, Tome 32 (2020) no. 4, pp. 166-199.

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A notion of almost minimizers is introduced for certain variational problems governed by the fractional Laplacian, with the help of the Caffarelli–Silvestre extension. In particular, almost fractional harmonic functions and almost minimizers for the fractional obstacle problem with zero obstacle are treated. It is shown that for a certain range of parameters, almost minimizers are almost Lipschitz or $C^{1,\beta}$-regular.
Keywords: almost minimizers, fractional Laplacian, fractional harmonic functions, fractional obstacle problem, regularity of solutions. 
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S. Jeon; A. Petrosyan. Almost minimizers for certain fractional variational problems. Algebra i analiz, Tome 32 (2020) no. 4, pp. 166-199. http://geodesic.mathdoc.fr/item/AA_2020_32_4_a3/

[1] Anzellotti G., “On the $C^{1,\alpha }$-regularity of $\omega $-minima of quadratic functionals”, Boll. Un. Mat. Ital. C (6), 2:1 (1983), 195–212 | MR | Zbl

[2] Axler Sh., Bourdon P., Ramey W., Harmonic function theory, Grad. Texts in Math., 137, Springer-Verlag, New York, 2001 | DOI | MR | Zbl

[3] Caffarelli L., Silvestre L., “An extension problem related to the fractional Laplacian”, Comm. Partial Differential Equations, 32:7-9 (2007), 1245–1260 | DOI | MR | Zbl

[4] 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., 171:2 (2008), 425–461 | DOI | MR | Zbl

[5] Fabes E. B., Kenig C. E., Serapioni R. P., “The local regularity of solutions of degenerate elliptic equations”, Comm. Partial Differential Equations, 7:1 (1982), 77–116 | DOI | MR | Zbl

[6] Franchi B., Serapioni R., “Pointwise estimates for a class of strongly degenerate elliptic operators: a geometrical approach”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 14:4 (1987), 527–568 | MR | Zbl

[7] Garofalo N., “Fractional thoughts”, New developments in the analysis of nonlocal operators, Contemp. Math., 723, Amer. Math. Soc., Providence, RI, 2019, 1–13 | DOI | MR

[8] Garofalo N., Petrosyan A., Pop C. A., Smit Vega Garcia M., “Regularity of the free boundary for the obstacle problem for the fractional Laplacian with drift”, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34:3 (2017), 533–570 | DOI | MR | Zbl

[9] Han Q., Lin F., Elliptic partial differential equations, Courant Lecture Notes in Math., 1, Amer. Math. Soc., New York–Providence, RI, 1997 | MR

[10] Jeon S., Petrosyan A., Almost minimizers for the thin obstacle problem, preprint, 2019

[11] Musina R., Nazarov A. I., Sreenadh K., “Variational inequalities for the fractional Laplacian”, Potential Anal., 46:3 (2017), 485–498 | DOI | 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., 268:2 (2015), 417–472 | DOI | MR | Zbl

[13] Petrosyan A., Shahgholian H., Uraltseva N., Regularity of free boundaries in obstacle-type problems, Grad. Stud. in Math., 136, Amer. Math. Soc., Providence, RI, 2012 | DOI | MR | Zbl

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