Graphical solutions to one-phase free boundary problems

References

[1] G. Alberti, L. Ambrosio and X. Cabré, On a long-standing conjecture of E. De Giorgi: Symmetry in 3D for general nonlinearities and a local minimality property, Acta Appl. Math. 65 (2001), 9–33. 10.1023/A:1010602715526Suche in Google Scholar

[2] M. Allen, Separation of a lower dimensional free boundary in a two-phase problem, Math. Res. Lett. 19 (2012), no. 5, 1055–1074. 10.4310/MRL.2012.v19.n5.a8Suche in Google Scholar

[3] F. J. Almgren, Jr., Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem, Ann. of Math. (2) 84 (1966), 277–292. 10.2307/1970520Suche in Google Scholar

[4] H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. reine angew. Math. 325 (1981), 105–144. 10.1515/crll.1981.325.105Suche in Google Scholar

[5] H. W. Alt, L. A. Caffarelli and A. Friedman, Asymmetric jet flows, Comm. Pure Appl. Math. 35 (1982), no. 1, 29–68. 10.1002/cpa.3160350103Suche in Google Scholar

[6] H. W. Alt, L. A. Caffarelli and A. Friedman, Jet flows with gravity, J. reine angew. Math. 331 (1982), 58–103. 10.1515/crll.1982.331.58Suche in Google Scholar

[7] H. W. Alt, L. A. Caffarelli and A. Friedman, Axially symmetric jet flows, Arch. Ration. Mech. Anal. 81 (1983), no. 2, 97–149. 10.1007/BF00250648Suche in Google Scholar

[8] L. Ambrosio and X. Cabré, Entire solutions of semilinear elliptic equations in 𝐑 3 and a conjecture of De Giorgi, J. Amer. Math. Soc. 13 (2000), no. 4, 725–739. 10.1090/S0894-0347-00-00345-3Suche in Google Scholar

[9] A. Audrito and J. Serra, Interface regularity for semilinear one-phase problems, Adv. Math. 403 (2022), Paper No. 108380. 10.1016/j.aim.2022.108380Suche in Google Scholar

[10] G. Baker, P. Saffman and J. Sheffield, Structure of a linear array of hollow vortices of finite cross-section, J. Fluid Mech. 74 (1976), 469–476. 10.1017/S0022112076001894Suche in Google Scholar

[11] H. Berestycki, L. A. Caffarelli and L. Nirenberg, Uniform estimates for regularization of free boundary problems, Analysis and partial differential equations, Lecture Notes Pure Appl. Math. 122, Dekker, New York (1990), 567–619. Suche in Google Scholar

[12] S. N. Bernstein, Sur une théorème de géometrie et ses applications aux équations dŕivées partielles du type elliptique, Comm. Soc. Math. Kharkov 15 (1915–1917), 38–45. Suche in Google Scholar

[13] E. Bombieri, E. De Giorgi and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243–268. 10.1007/BF01404309Suche in Google Scholar

[14] E. Bombieri and E. Giusti, Harnack’s inequality for elliptic differential equations on minimal surfaces, Invent. Math. 15 (1972), 24–46. 10.1007/BF01418640Suche in Google Scholar

[15] J. D. Buckmaster and G. S. S. Ludford, Theory of laminar flames, Cambridge University, New York 1982. 10.1017/CBO9780511569531Suche in Google Scholar

[16] X. Cabré, I. U. Erneta and J. C. Felipe-Navarro, A Weierstrass extremal field theory for the fractional Laplacian, preprint (2022), https://arxiv.org/abs/2211.16536. Suche in Google Scholar

[17] X. Cabré and G. Poggesi, Stable solutions to some elliptic problems: Minimal cones, the Allen–Cahn equation, and blow-up solutions, Geometry of PDEs and related problems, Lecture Notes in Math. 2220, Springer, Cham (2018), 1–45. 10.1007/978-3-319-95186-7_1Suche in Google Scholar

[18] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré C Anal. Non Linéaire 31 (2014), no. 1, 23–53. 10.1016/j.anihpc.2013.02.001Suche in Google Scholar

[19] L. Caffarelli and S. Salsa, A geometric approach to free boundary problems, Grad. Stud. Math. 68, American Mathematical Society, Providence 2005. 10.1090/gsm/068Suche in Google Scholar

[20] L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. I. Lipschitz free boundaries are C 1 , α , Rev. Mat. Iberoam. 3 (1987), no. 2, 139–162. 10.4171/RMI/47Suche in Google Scholar

[21] L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. III. Existence theory, compactness, and dependence on X, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 15 (1988), no. 4, 583–602. Suche in Google Scholar

[22] L. A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. II. Flat free boundaries are Lipschitz, Comm. Pure Appl. Math. 42 (1989), no. 1, 55–78. 10.1002/cpa.3160420105Suche in Google Scholar

[23] L. A. Caffarelli, D. Jerison and C. E. Kenig, Global energy minimizers for free boundary problems and full regularity in three dimensions, Noncompact problems at the intersection of geometry, analysis, and topology, Contemp. Math. 350, American Mathematical Society, Providence (2004), 83–97. 10.1090/conm/350/06339Suche in Google Scholar

[24] L. A. Caffarelli, J.-M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 5, 1151–1179. 10.4171/JEMS/226Suche in Google Scholar

[25] L. A. Caffarelli and J. L. Vázquez, A free-boundary problem for the heat equation arising in flame propagation, Trans. Amer. Math. Soc. 347 (1995), no. 2, 411–441. 10.1090/S0002-9947-1995-1260199-7Suche in Google Scholar

[26] H. Chang-Lara and O. Savin, Boundary regularity for the free boundary in the one-phase problem, New developments in the analysis of nonlocal operators, Contemp. Math. 723, American Mathematical Society, Providence (2019), 149–165. 10.1090/conm/723/14549Suche in Google Scholar

[27] T. H. Colding and W. P. Minicozzi, II, A course in minimal surfaces, Grad. Stud. Math. 121, American Mathematical Society, Providence 2011. 10.1090/gsm/121Suche in Google Scholar

[28] D. Crowdy and C. Green, Analytical solutions for von Kármán streets of hollow vortices, Phys. Fluids 23 (2011), Article ID 126602. 10.1063/1.3665102Suche in Google Scholar

[29] G. David, M. Engelstein and T. Toro, Free boundary regularity for almost-minimizers, Adv. Math. 350 (2019), 1109–1192. 10.1016/j.aim.2019.04.059Suche in Google Scholar

[30] G. David and T. Toro, Regularity of almost minimizers with free boundary, Calc. Var. Partial Differential Equations 54 (2015), no. 1, 455–524. 10.1007/s00526-014-0792-zSuche in Google Scholar

[31] E. De Giorgi, Una estensione del teorema di Bernstein, Ann. Sc. Norm. Super. Pisa Cl. Sci. (3) 19 (1965), 79–85. Suche in Google Scholar

[32] E. De Giorgi, Convergence problems for functionals and operators, Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis, Pitagora, Bologna (1979), 131–188. Suche in Google Scholar

[33] D. De Silva, Free boundary regularity for a problem with right hand side, Interfaces Free Bound. 13 (2011), no. 2, 223–238. 10.4171/IFB/255Suche in Google Scholar

[34] D. De Silva and D. Jerison, A singular energy minimizing free boundary, J. reine angew. Math. 635 (2009), 1–21. 10.1515/CRELLE.2009.074Suche in Google Scholar

[35] D. De Silva and D. Jerison, A gradient bound for free boundary graphs, Comm. Pure Appl. Math. 64 (2011), no. 4, 538–555. 10.1002/cpa.20354Suche in Google Scholar

[36] D. De Silva, D. Jerison and H. Shahgholian, Inhomogeneous global minimizers to the one-phase free boundary problem, Comm. Partial Differential Equations 47 (2022), no. 6, 1193–1216. 10.1080/03605302.2022.2051187Suche in Google Scholar

[37] D. De Silva and J. M. Roquejoffre, Regularity in a one-phase free boundary problem for the fractional Laplacian, Ann. Inst. H. Poincaré C Anal. Non Linéaire 29 (2012), no. 3, 335–367. 10.1016/j.anihpc.2011.11.003Suche in Google Scholar

[38] D. De Silva and O. Savin, C 2 , α regularity of flat free boundaries for the thin one-phase problem, J. Differential Equations 253 (2012), no. 8, 2420–2459. 10.1016/j.jde.2012.06.021Suche in Google Scholar

[39] D. De Silva and O. Savin, C ∞ regularity of certain thin free boundaries, Indiana Univ. Math. J. 64 (2015), no. 5, 1575–1608. 10.1512/iumj.2015.64.5632Suche in Google Scholar

[40] D. De Silva and O. Savin, Regularity of Lipschitz free boundaries for the thin one-phase problem, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 6, 1293–1326. 10.4171/JEMS/531Suche in Google Scholar

[41] D. De Silva and O. Savin, A short proof of boundary Harnack principle, J. Differential Equations 269 (2020), no. 3, 2419–2429. 10.1016/j.jde.2020.02.004Suche in Google Scholar

[42] D. De Silva, O. Savin and Y. Sire, A one-phase problem for the fractional Laplacian: Regularity of flat free boundaries, Bull. Inst. Math. Acad. Sin. (N. S.) 9 (2014), no. 1, 111–145. Suche in Google Scholar

[43] M. del Pino, M. Kowalczyk and J. Wei, On De Giorgi’s conjecture in dimension N ≥ 9 , Ann. of Math. (2) 174 (2011), no. 3, 1485–1569. 10.4007/annals.2011.174.3.3Suche in Google Scholar

[44] S. Eberle, A. Figalli and G. Weiss, Complete classification of global solutions to the obstacle problem, preprint (2022), https://arxiv.org/abs/2208.03108. Suche in Google Scholar

[45] S. Eberle, X. Ros-Oton and G. S. Weiss, Characterizing compact coincidence sets in the thin obstacle problem and the obstacle problem for the fractional Laplacian, Nonlinear Anal. 211 (2021), Paper No. 112473. 10.1016/j.na.2021.112473Suche in Google Scholar

[46] S. Eberle, H. Shahgholian and G. Weiss, The structure of the regular part of the free boundary close to singularities in the obstacle problem, preprint (2022), https://arxiv.org/abs/2204.03337; to appear in J. Differential Equations. 10.2139/ssrn.4296060Suche in Google Scholar

[47] S. Eberle, H. Shahgholian and G. S. Weiss, On global solutions of the obstacle problem, Duke Math. J. 172 (2023), no. 11, 2149–2193. 10.1215/00127094-2022-0078Suche in Google Scholar

[48] S. Eberle and H. Yu, Compact compact sets of sub-quadratic solutions to the thin obstacle problem, preprint (2023), https://arxiv.org/abs/2304.03939. Suche in Google Scholar

[49] S. Eberle and H. Yu, Solutions to the nonlinear obstacle problem with compact contact sets, preprint (2023), https://arxiv.org/abs/2305.19963. Suche in Google Scholar

[50] N. Edelen and M. Engelstein, Quantitative stratification for some free-boundary problems, Trans. Amer. Math. Soc. 371 (2019), no. 3, 2043–2072. 10.1090/tran/7401Suche in Google Scholar

[51] N. Edelen, L. Spolaor and B. Velichkov, A strong maximum principle for minimizers of the one-phase Bernoulli problem, preprint (2022), https://arxiv.org/abs/2205.00401; to appear in Indiana Univ. Math. J. Suche in Google Scholar

[52] M. Engelstein, A. Kauranen, M. Prats, G. Sakellaris and Y. Sire, Minimizers for the thin one-phase free boundary problem, Comm. Pure Appl. Math. 74 (2021), no. 9, 1971–2022. 10.1002/cpa.22011Suche in Google Scholar

[53] M. Engelstein, L. Spolaor and B. Velichkov, Uniqueness of the blowup at isolated singularities for the Alt–Caffarelli functional, Duke Math. J. 169 (2020), no. 8, 1541–1601. 10.1215/00127094-2019-0077Suche in Google Scholar

[54] X. Fernández-Real and X. Ros-Oton, On global solutions to semilinear elliptic equations related to the one-phase free boundary problem, Discrete Contin. Dyn. Syst. 39 (2019), no. 12, 6945–6959. 10.3934/dcds.2019238Suche in Google Scholar

[55] X. Fernández-Real and X. Ros-Oton, Regularity theory for elliptic PDE, Zur. Lect. Adv. Math. 28, EMS Press, Berlin 2022. 10.4171/zlam/28Suche in Google Scholar

[56] X. Fernández-Real and X. Ros-Oton, Stable cones in the thin one-phase problem, preprint (2022), https://arxiv.org/abs/2009.11626; to appear in Amer. J. Math Suche in Google Scholar

[57] X. Fernández-Real and H. Yu, Generic properties in free boundary problems, preprint (2023), https://arxiv.org/abs/2308.13209. Suche in Google Scholar

[58] W. H. Fleming, On the oriented Plateau problem, Rend. Circ. Mat. Palermo (2) 11 (1962), 69–90. 10.1007/BF02849427Suche in Google Scholar

[59] N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann. 311 (1998), no. 3, 481–491. 10.1007/s002080050196Suche in Google Scholar

[60] L. Hauswirth, F. Hélein and F. Pacard, On an overdetermined elliptic problem, Pacific J. Math. 250 (2011), no. 2, 319–334. 10.2140/pjm.2011.250.319Suche in Google Scholar

[61] D. Jerison and N. Kamburov, Structure of one-phase free boundaries in the plane, Int. Math. Res. Not. IMRN 2016 (2016), no. 19, 5922–5987. 10.1093/imrn/rnv339Suche in Google Scholar

[62] D. Jerison and O. Savin, Some remarks on stability of cones for the one-phase free boundary problem, Geom. Funct. Anal. 25 (2015), no. 4, 1240–1257. 10.1007/s00039-015-0335-6Suche in Google Scholar

[63] N. Kamburov and K. Wang, Nondegeneracy for stable solutions to the one-phase free boundary problem, preprint (2022), https://arxiv.org/abs/2207.12740; to appear in Math. Ann. 10.1007/s00208-023-02591-0Suche in Google Scholar

[64] D. Khavinson, E. Lundberg and R. Teodorescu, An overdetermined problem in potential theory, Pacific J. Math. 265 (2013), no. 1, 85–111. 10.2140/pjm.2013.265.85Suche in Google Scholar

[65] Y. Liu, K. Wang and J. Wei, On smooth solutions to one phase-free boundary problem in ℝ n , Int. Math. Res. Not. IMRN 2021 (2021), no. 20, 15682–15732. 10.1093/imrn/rnz250Suche in Google Scholar

[66] X. Ros-Oton and C. Torres-Latorre, New boundary Harnack inequalities with right hand side, J. Differential Equations 288 (2021), 204–249. 10.1016/j.jde.2021.04.012Suche in Google Scholar

[67] O. Savin, Regularity of flat level sets in phase transitions, Ann. of Math. (2) 169 (2009), no. 1, 41–78. 10.4007/annals.2009.169.41Suche in Google Scholar

[68] J. Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. 10.2307/1970556Suche in Google Scholar

[69] M. Traizet, Classification of the solutions to an overdetermined elliptic problem in the plane, Geom. Funct. Anal. 24 (2014), no. 2, 690–720. 10.1007/s00039-014-0268-5Suche in Google Scholar

[70] B. Velichkov, Regularity of the one-phase free boundaries, Lect. Notes Unione Mat. Ital. 28, Springer, Cham 2023. 10.1007/978-3-031-13238-4Suche in Google Scholar

[71] G. S. Weiss, Partial regularity for a minimum problem with free boundary, J. Geom. Anal. 9 (1999), no. 2, 317–326. 10.1007/BF02921941Suche in Google Scholar