Continuity and density results for a one-phase nonlocal free boundary problem
Annales de l'I.H.P. Analyse non linéaire, Volume 34 (2017) no. 6, pp. 1387-1428.

We consider a one-phase nonlocal free boundary problem obtained by the superposition of a fractional Dirichlet energy plus a nonlocal perimeter functional. We prove that the minimizers are Hölder continuous and the free boundary has positive density from both sides.

For this, we also introduce a new notion of fractional harmonic replacement in the extended variables and we study its basic properties.

DOI: 10.1016/j.anihpc.2016.11.001
Classification: 35R35, 49N60, 35R11, 35A15
Keywords: Free boundary problems, Nonlocal minimal surfaces, Fractional operators, Regularity theory, Fractional harmonic replacement
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Dipierro, Serena; Valdinoci, Enrico. Continuity and density results for a one-phase nonlocal free boundary problem. Annales de l'I.H.P. Analyse non linéaire, Volume 34 (2017) no. 6, pp. 1387-1428. doi : 10.1016/j.anihpc.2016.11.001. http://www.numdam.org/articles/10.1016/j.anihpc.2016.11.001/

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

[2] Alt, H.W.; Caffarelli, L.A.; Friedman, A. Variational problems with two phases and their free boundary, Trans. Am. Math. Soc., Volume 282 (1984) no. 2, pp. 431–461 | MR | Zbl

[3] Athanasopoulos, I.; Caffarelli, L.A.; Kenig, C.; Salsa, S. An area-Dirichlet integral minimization problem, Commun. Pure Appl. Math., Volume 54 (2001) no. 4, pp. 479–499 | DOI | MR | Zbl

[4] Brezis, H. Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011 (xiv, 599 pp.) | MR | Zbl

[5] Caffarelli, L.; Roquejoffre, J.-M.; Savin, O. Nonlocal minimal surfaces, Commun. Pure Appl. Math., Volume 63 (2010) no. 9, pp. 1111–1144 | DOI | MR | Zbl

[6] Caffarelli, L.; Savin, O.; Valdinoci, E. Minimization of a fractional perimeter-Dirichlet integral functional, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 32 (2015) no. 4, pp. 901–924 | DOI | Numdam | MR | Zbl

[7] Caffarelli, L.; 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

[8] Caffarelli, L.; Silvestre, L. Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., Volume 200 (2011) no. 1, pp. 59–88 | DOI | MR | Zbl

[9] Capriani, G.M. The Steiner rearrangement in any codimension, Calc. Var. Partial Differ. Equ., Volume 49 (2014) no. 1–2, pp. 517–548 | MR | Zbl

[10] Di Nezza, E.; Palatucci, G.; Valdinoci, E. Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., Volume 136 (2012) no. 5, pp. 521–573 | DOI | MR | Zbl

[11] Dipierro, S.; Figalli, A.; Valdinoci, E. Strongly nonlocal dislocation dynamics in crystals, Commun. Partial Differ. Equ., Volume 39 (2014) no. 12, pp. 2351–2387 | DOI | MR | Zbl

[12] Dipierro, S.; Medina, M.; Valdinoci, E. Fractional elliptic problems with critical growth in the whole of Rn (preprint) | arXiv | MR

[13] Dipierro, S.; Savin, O.; Valdinoci, E. All functions are locally s-harmonic up to a small error, J. Eur. Math. Soc. (JEMS) (2016) (in press) | MR

[14] Dipierro, S.; Savin, O.; Valdinoci, E. A nonlocal free boundary problem, SIAM J. Math. Anal., Volume 47 (2015) no. 6, pp. 4559–4605 | DOI | MR | Zbl

[15] Dipierro, S.; Savin, O.; Valdinoci, E. Definition of fractional Laplacian for functions with polynomial growth (preprint) | arXiv | DOI | MR | Zbl

[16] Dipierro, S.; Valdinoci, E. On a fractional harmonic replacement, Discrete Contin. Dyn. Syst., Volume 35 (2015) no. 8, pp. 3377–3392 | DOI | MR | Zbl

[17] dos Prazeres, D.; Teixeira, E.V. Cavity problems in discontinuous media, Calc. Var. Partial Differ. Equ., Volume 55 (2016) no. 1 | DOI | MR | Zbl

[18] 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

[19] Heinonen, J.; Kilpeläinen, T.; Martio, O. Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1993 (v, 363 pp.) | MR | Zbl

[20] Kassmann, M. A new formulation of Harnack's inequality for nonlocal operators, C. R. Math. Acad. Sci. Paris, Volume 349 (2011) no. 11–12, pp. 637–640 | MR | Zbl

[21] Lieb, E.H.; Loss, M. Analysis, Graduate Studies in Mathematics, vol. 14, American Mathematical Society (AMS), Providence, RI, 2001 (xxii, 346 pp.) | MR | Zbl

[22] Teixeira, E.V.; Zhang, L. An elliptic variational problem involving level surface area on Riemannian manifolds, Rev. Mat. Iberoam., Volume 28 (2012) no. 3, pp. 759–772 | DOI | MR | Zbl

[23] Tyulenev, A.I. Boundary values of functions in a Sobolev space with Muckenhoupt weight on some non-Lipschitz domains, Sb. Math., Volume 205 (2014) no. 8, pp. 1133–1159 | DOI | MR | Zbl

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