Global weighted estimates for nonlinear elliptic obstacle problems over Reifenberg domains. (English) Zbl 1315.35113
Summary: We study the obstacle problem for an elliptic equation with discontinuous nonlinearity over a nonsmooth domain, assuming that the irregular obstacle and the nonhomogeneous term belong to suitable weighted Sobolev and Lebesgue spaces, respectively, with weights taken in the Muckenhoupt classes. We establish a Calderón-Zygmund type result by proving that the gradient of the weak solution to the nonlinear obstacle problem has the same weighted integrability as both the gradient of the obstacle and the nonhomogeneous term, provided that the nonlinearity has a small BMO-semi norm with respect to the gradient, and the boundary of the domain is \( \delta \)-Reifenberg flat. We also get global regularity in the settings of the Morrey and Hölder spaces for the weak solutions to the problem considered.
MSC:
| 35J87 | Unilateral problems for nonlinear elliptic equations and variational inequalities with nonlinear elliptic operators |
| 35R05 | PDEs with low regular coefficients and/or low regular data |
| 35J60 | Nonlinear elliptic equations |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 35D30 | Weak solutions to PDEs |
Keywords:
nonlinear elliptic equation; obstacle problem; irregular obstacle; Calderón-Zygmund estimate; Muckenhoupt weight; \(p\)-Laplacean; BMO; Reifenberg flat domain; Morrey spaceReferences:
| [1] | B{\"o}gelein, Verena; Duzaar, Frank; Mingione, Giuseppe, Degenerate problems with irregular obstacles, J. Reine Angew. Math., 650, 107-160 (2011) · Zbl 1218.35088 · doi:10.1515/CRELLE.2011.006 |
| [2] | B{\"o}gelein, Verena; Scheven, Christoph, Higher integrability in parabolic obstacle problems, Forum Math., 24, 5, 931-972 (2012) · Zbl 1262.35140 · doi:10.1515/form.2011.091 |
| [3] | Byun, Sun-Sig; Palagachev, Dian K., Morrey regularity of solutions to quasilinear elliptic equations over Reifenberg flat domains, Calc. Var. Partial Differential Equations, 49, 1-2, 37-76 (2014) · Zbl 1288.35255 · doi:10.1007/s00526-012-0574-4 |
| [4] | [BPiumj] S. Byun and D.K. Palagachev, Boundedness of the weak solutions to quasilinear elliptic equations with Morrey data, Indiana Univ. Math. J. 62 (2013), no. 5, 1565-1585. · Zbl 1300.35045 |
| [5] | Byun, Sun-Sig; Palagachev, Dian K.; Ryu, Seungjin, Weighted \(W^{1,p}\) estimates for solutions of non-linear parabolic equations over non-smooth domains, Bull. Lond. Math. Soc., 45, 4, 765-778 (2013) · Zbl 1317.35115 · doi:10.1112/blms/bdt011 |
| [6] | Byun, Sun-Sig; Cho, Yumi; Wang, Lihe, Calder\'on-Zygmund theory for nonlinear elliptic problems with irregular obstacles, J. Funct. Anal., 263, 10, 3117-3143 (2012) · Zbl 1259.35094 · doi:10.1016/j.jfa.2012.07.018 |
| [7] | Byun, Sun-Sig; Wang, Lihe, Parabolic equations in time dependent Reifenberg domains, Adv. Math., 212, 2, 797-818 (2007) · Zbl 1117.35080 · doi:10.1016/j.aim.2006.12.002 |
| [8] | Byun, Sun-Sig; Wang, Lihe, Nonlinear gradient estimates for elliptic equations of general type, Calc. Var. Partial Differential Equations, 45, 3-4, 403-419 (2012) · Zbl 1263.35102 · doi:10.1007/s00526-011-0463-2 |
| [9] | Caffarelli, L. A., The obstacle problem revisited, J. Fourier Anal. Appl., 4, 4-5, 383-402 (1998) · Zbl 0928.49030 · doi:10.1007/BF02498216 |
| [10] | Campanato, Sergio, Propriet\`a di inclusione per spazi di Morrey, Ricerche Mat., 12, 67-86 (1963) · Zbl 0192.22703 |
| [11] | Coifman, R. R.; Rochberg, R., Another characterization of BMO, Proc. Amer. Math. Soc., 79, 2, 249-254 (1980) · Zbl 0432.42016 · doi:10.2307/2043245 |
| [12] | David, Guy; Toro, Tatiana, Reifenberg parameterizations for sets with holes, Mem. Amer. Math. Soc., 215, 1012, vi+102 pp. (2012) · Zbl 1236.28002 · doi:10.1090/S0065-9266-2011-00629-5 |
| [13] | Eleuteri, Michela; Habermann, Jens, Calder\'on-Zygmund type estimates for a class of obstacle problems with \(p(x)\) growth, J. Math. Anal. Appl., 372, 1, 140-161 (2010) · Zbl 1211.49046 · doi:10.1016/j.jmaa.2010.05.072 |
| [14] | Evans, Lawrence C., Partial differential equations, Graduate Studies in Mathematics 19, xxii+749 pp. (2010), American Mathematical Society: Providence, RI:American Mathematical Society · Zbl 1194.35001 |
| [15] | Friedman, Avner, Variational principles and free-boundary problems, Pure and Applied Mathematics, ix+710 pp. (1982), John Wiley & Sons Inc.: New York:John Wiley & Sons Inc. · Zbl 0671.49001 |
| [16] | Kenig, Carlos E.; Toro, Tatiana, Free boundary regularity for harmonic measures and Poisson kernels, Ann. of Math. (2), 150, 2, 369-454 (1999) · Zbl 0946.31001 · doi:10.2307/121086 |
| [17] | Kinderlehrer, David; Stampacchia, Guido, An introduction to variational inequalities and their applications, Classics in Applied Mathematics 31, xx+313 pp. (2000), Society for Industrial and Applied Mathematics (SIAM): Philadelphia, PA:Society for Industrial and Applied Mathematics (SIAM) · Zbl 0988.49003 · doi:10.1137/1.9780898719451 |
| [18] | Ladyzhenskaya, O. A.; Uraltseva, N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, 576 pp. (1973), Izdat. “Nauka”, Moscow · Zbl 0269.35029 |
| [19] | [LMS] A. Lemenant, E. Milakis and L. V. Spinolo, On the extension property of Reifenberg-flat domains, (2012), arXiv:1209.3602. · Zbl 1292.49042 |
| [20] | Lewis, John L.; Nystr{\"o}m, Kaj, Regularity and free boundary regularity for the \(p\)-Laplace operator in Reifenberg flat and Ahlfors regular domains, J. Amer. Math. Soc., 25, 3, 827-862 (2012) · Zbl 1250.35084 · doi:10.1090/S0894-0347-2011-00726-1 |
| [21] | Mengesha, Tadele; Phuc, Nguyen Cong, Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains, J. Differential Equations, 250, 5, 2485-2507 (2011) · Zbl 1210.35094 · doi:10.1016/j.jde.2010.11.009 |
| [22] | Muckenhoupt, Benjamin, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165, 207-226 (1972) · Zbl 0236.26016 |
| [23] | Palagachev, Dian K.; Softova, Lubomira G., Quasilinear divergence form parabolic equations in Reifenberg flat domains, Discrete Contin. Dyn. Syst., 31, 4, 1397-1410 (2011) · Zbl 1235.35176 · doi:10.3934/dcds.2011.31.1397 |
| [24] | Palagachev, Dian K.; Softova, Lubomira G., The Calder\'on-Zygmund property for quasilinear divergence form equations over Reifenberg flat domains, Nonlinear Anal., 74, 5, 1721-1730 (2011) · Zbl 1209.35047 · doi:10.1016/j.na.2010.10.044 |
| [25] | Reifenberg, E. R., Solution of the Plateau Problem for \(m\)-dimensional surfaces of varying topological type, Acta Math., 104, 1-92 (1960) · Zbl 0099.08503 |
| [26] | Rodrigues, Jos{\'e}-Francisco, Obstacle problems in mathematical physics, North-Holland Mathematics Studies 134, xvi+352 pp. (1987), North-Holland Publishing Co.: Amsterdam:North-Holland Publishing Co. · Zbl 0606.73017 |
| [27] | [Sch] C. Scheven, Existence of localizable solutions to nonlinear parabolic problems with irregular obstacles, preprint, (2011). |
| [28] | Scheven, Christoph, Gradient potential estimates in non-linear elliptic obstacle problems with measure data, J. Funct. Anal., 262, 6, 2777-2832 (2012) · Zbl 1238.35030 · doi:10.1016/j.jfa.2012.01.003 |
| [29] | Softova, Lubomira, Singular integrals and commutators in generalized Morrey spaces, Acta Math. Sin. (Engl. Ser.), 22, 3, 757-766 (2006) · Zbl 1129.42372 · doi:10.1007/s10114-005-0628-z |
| [30] | Torchinsky, Alberto, Real-variable methods in harmonic analysis, Pure and Applied Mathematics 123, xii+462 pp. (1986), Academic Press Inc.: Orlando, FL:Academic Press Inc. · Zbl 1097.42002 |
| [31] | Toro, Tatiana, Doubling and flatness: geometry of measures, Notices Amer. Math. Soc., 44, 9, 1087-1094 (1997) · Zbl 0909.31006 |
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