Global weighted estimates for nonlinear elliptic obstacle problems over Reifenberg domains

Byun, Sun-Sig; Cho, Yumi; Palagachev, Dian

doi:10.1090/S0002-9939-2015-12458-6

Global weighted estimates for nonlinear elliptic obstacle problems over Reifenberg domains
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by Sun-Sig Byun, Yumi Cho and Dian K. Palagachev

Proc. Amer. Math. Soc. 143 (2015), 2527-2541

DOI: https://doi.org/10.1090/S0002-9939-2015-12458-6

Published electronically: January 21, 2015

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Abstract:

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.

References

Verena Bögelein, Frank Duzaar, and Giuseppe Mingione, Degenerate problems with irregular obstacles, J. Reine Angew. Math. 650 (2011), 107–160. MR 2770559, DOI 10.1515/CRELLE.2011.006
Verena Bögelein and Christoph Scheven, Higher integrability in parabolic obstacle problems, Forum Math. 24 (2012), no. 5, 931–972. MR 2988568, DOI 10.1515/form.2011.091
Sun-Sig Byun and Dian K. Palagachev, Morrey regularity of solutions to quasilinear elliptic equations over Reifenberg flat domains, Calc. Var. Partial Differential Equations 49 (2014), no. 1-2, 37–76. MR 3148106, DOI 10.1007/s00526-012-0574-4
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.
Sun-Sig Byun, Dian K. Palagachev, and Seungjin Ryu, Weighted $W^{1,p}$ estimates for solutions of non-linear parabolic equations over non-smooth domains, Bull. Lond. Math. Soc. 45 (2013), no. 4, 765–778. MR 3081545, DOI 10.1112/blms/bdt011
Sun-Sig Byun, Yumi Cho, and Lihe Wang, Calderón-Zygmund theory for nonlinear elliptic problems with irregular obstacles, J. Funct. Anal. 263 (2012), no. 10, 3117–3143. MR 2973336, DOI 10.1016/j.jfa.2012.07.018
Sun-Sig Byun and Lihe Wang, Parabolic equations in time dependent Reifenberg domains, Adv. Math. 212 (2007), no. 2, 797–818. MR 2329320, DOI 10.1016/j.aim.2006.12.002
Sun-Sig Byun and Lihe Wang, Nonlinear gradient estimates for elliptic equations of general type, Calc. Var. Partial Differential Equations 45 (2012), no. 3-4, 403–419. MR 2984138, DOI 10.1007/s00526-011-0463-2
L. A. Caffarelli, The obstacle problem revisited, J. Fourier Anal. Appl. 4 (1998), no. 4-5, 383–402. MR 1658612, DOI 10.1007/BF02498216
Sergio Campanato, Proprietà di inclusione per spazi di Morrey, Ricerche Mat. 12 (1963), 67–86 (Italian). MR 156228
R. R. Coifman and R. Rochberg, Another characterization of BMO, Proc. Amer. Math. Soc. 79 (1980), no. 2, 249–254. MR 565349, DOI 10.1090/S0002-9939-1980-0565349-8
Guy David and Tatiana Toro, Reifenberg parameterizations for sets with holes, Mem. Amer. Math. Soc. 215 (2012), no. 1012, vi+102. MR 2907827, DOI 10.1090/S0065-9266-2011-00629-5
Michela Eleuteri and Jens Habermann, Calderón-Zygmund type estimates for a class of obstacle problems with $p(x)$ growth, J. Math. Anal. Appl. 372 (2010), no. 1, 140–161. MR 2672516, DOI 10.1016/j.jmaa.2010.05.072
Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943, DOI 10.1090/gsm/019
Avner Friedman, Variational principles and free-boundary problems, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1982. MR 679313
Carlos E. Kenig and Tatiana Toro, Free boundary regularity for harmonic measures and Poisson kernels, Ann. of Math. (2) 150 (1999), no. 2, 369–454. MR 1726699, DOI 10.2307/121086
David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Classics in Applied Mathematics, vol. 31, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. Reprint of the 1980 original. MR 1786735, DOI 10.1137/1.9780898719451
O. A. Ladyzhenskaya and N. N. Ural′tseva, Lineĭnye i kvazilineĭnye uravneniya èllipticheskogo tipa, Izdat. “Nauka”, Moscow, 1973 (Russian). Second edition, revised. MR 0509265
A. Lemenant, E. Milakis and L. V. Spinolo, On the extension property of Reifenberg-flat domains, (2012), arXiv:1209.3602.
John L. Lewis and Kaj Nyström, Regularity and free boundary regularity for the $p$-Laplace operator in Reifenberg flat and Ahlfors regular domains, J. Amer. Math. Soc. 25 (2012), no. 3, 827–862. MR 2904575, DOI 10.1090/S0894-0347-2011-00726-1
Tadele Mengesha and Nguyen Cong Phuc, Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains, J. Differential Equations 250 (2011), no. 5, 2485–2507. MR 2756073, DOI 10.1016/j.jde.2010.11.009
Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. MR 293384, DOI 10.1090/S0002-9947-1972-0293384-6
Dian K. Palagachev and Lubomira G. Softova, Quasilinear divergence form parabolic equations in Reifenberg flat domains, Discrete Contin. Dyn. Syst. 31 (2011), no. 4, 1397–1410. MR 2836359, DOI 10.3934/dcds.2011.31.1397
Dian K. Palagachev and Lubomira G. Softova, The Calderón-Zygmund property for quasilinear divergence form equations over Reifenberg flat domains, Nonlinear Anal. 74 (2011), no. 5, 1721–1730. MR 2764374, DOI 10.1016/j.na.2010.10.044
E. R. Reifenberg, Solution of the Plateau Problem for $m$-dimensional surfaces of varying topological type, Acta Math. 104 (1960), 1–92. MR 114145, DOI 10.1007/BF02547186
José-Francisco Rodrigues, Obstacle problems in mathematical physics, North-Holland Mathematics Studies, vol. 134, North-Holland Publishing Co., Amsterdam, 1987. Notas de Matemática [Mathematical Notes], 114. MR 880369
C. Scheven, Existence of localizable solutions to nonlinear parabolic problems with irregular obstacles, preprint, (2011).
Christoph Scheven, Gradient potential estimates in non-linear elliptic obstacle problems with measure data, J. Funct. Anal. 262 (2012), no. 6, 2777–2832. MR 2885965, DOI 10.1016/j.jfa.2012.01.003
Lubomiea Softova, Singular integrals and commutators in generalized Morrey spaces, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 3, 757–766. MR 2219687, DOI 10.1007/s10114-005-0628-z
Alberto Torchinsky, Real-variable methods in harmonic analysis, Pure and Applied Mathematics, vol. 123, Academic Press, Inc., Orlando, FL, 1986. MR 869816
Tatiana Toro, Doubling and flatness: geometry of measures, Notices Amer. Math. Soc. 44 (1997), no. 9, 1087–1094. MR 1470167