Kim, Youchan;Ryu, Seungjin

  • Received : 2018.03.06
  • Accepted : 2018.07.05
  • Published : 2019.01.01


The $Calder{\acute{o}}n$-Zygmund type estimate is proved for elliptic obstacle problems in bounded non-smooth domains. The problems are related to divergence form nonlinear elliptic equation with measurable nonlinearities. Precisely, nonlinearity $a({\xi},x_1,x^{\prime})$ is assumed to be only measurable in one spatial variable $x_1$ and has locally small BMO semi-norm in the other spatial variables x', uniformly in ${\xi}$ variable. Regarding non-smooth domains, we assume that the boundaries are locally flat in the sense of Reifenberg. We also investigate global regularity in the settings of weighted Orlicz spaces for the weak solutions to the problems considered here.


$Calder{\acute{o}}n$-Zygmund type estimate;nonlinear elliptic obstacle problem;measurable nonlinearity;BMO;Reifenberg flat domain


  1. G. Bao, T. Wang, and G. Li, On very weak solutions to a class of double obstacle problems, J. Math. Anal. Appl. 402 (2013), no. 2, 702-709.
  2. P. Baroni, Lorentz estimates for obstacle parabolic problems, Nonlinear Anal. 96 (2014), 167-188.
  3. M. Bildhauer, M. Fuchs, and G. Mingione, A priori gradient bounds and local $C^{1,{\alpha}}$-estimates for (double) obstacle problems under non-standard growth conditions, Z. Anal. Anwendungen 20 (2001), no. 4, 959-985.
  4. T. A. Bui and X. T. Le, $W^{1,p({\cdot})}$ regularity for quasilinear problems with irregular obstacles on Reifenberg domains, Commun. Contemp. Math. 19 (2017), no. 6, 1650046, 19 pp.
  5. V. Bogelein, F. Duzzar, and G. Mingione, Degenerate problems with irregular obstacles, J. Reine Angew. Math. 650 (2011), 107-160.
  6. V. Bogelein and M. Parviaine, Self-improving property of nonlinear higher order parabolic systems near the boundary, NoDEA Nonlinear Differential Equations Appl. 17 (2010), no. 1, 21-54.
  7. S.-S. Byun, Y. Cho, and L. Wang, Calderon-Zygmund theory for nonlinear elliptic problems with irregular obstacles, J. Funct. Anal. 263 (2012), no. 10, 3117-3143.
  8. S.-S. Byun and Y. Kim, Elliptic equations with measurable nonlinearities in nonsmooth domains, Adv. Math. 288 (2016), 152-200.
  9. S.-S. Byun, J. Ok, D. K. Palagachev, and L. G. Softova, Parabolic systems with measurable coefficients in weighted Orlicz spaces, Commun. Contemp. Math. 18 (2016), no. 2, 1550018, 19 pp.
  10. S.-S. Byun, D. K. Palagachev, and S. Ryu, Elliptic obstacle problems with measurable coefficients in non-smooth domains, Numer. Funct. Anal. Optim. 35 (2014), no. 7-9, 893-910.
  11. S.-S. Byun and L. Softova, Parabolic obstacle problem with measurable data in generalized Morrey spaces, Z. Anal. Anwend. 35 (2016), no. 2, 153-171.
  12. H. J. Choe, A regularity theory for a general class of quasilinear elliptic partial differential equations and obstacle problems, Arch. Rational Mech. Anal. 114 (1991), no. 4, 383-394.
  13. H. J. Choe and J. L. Lewis, On the obstacle problem for quasilinear elliptic equations of p Laplacian type, SIAM J. Math. Anal. 22 (1991), no. 3, 623-638.
  14. H. Dong and D. Kim, Elliptic equations in divergence form with partially BMO coefficients, Arch. Ration. Mech. Anal. 196 (2010), no. 1, 25-70.
  15. H. Dong and D. Kim, Higher order elliptic and parabolic systems with variably partially BMO coefficients in regular and irregular domains, J. Funct. Anal. 261 (2011), no. 11, 3279-3327.
  16. H. Dong and D. Kim, $L_p$ solvability of divergence type parabolic and elliptic systems with partially BMO coefficients, Calc. Var. Partial Differential Equations 40 (2011), no. 3-4, 357-389.
  17. A. Fiorenza and M. Krbec, Indices of Orlicz spaces and some applications, Comment. Math. Univ. Carolin. 38 (1997), no. 3, 433-451.
  18. A. Friedman, Variational Principles and Free-Boundary Problems, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982.
  19. M. Fuchs and G. Mingione, Full $C^{1,{\alpha}}$-regularity for free and constrained local minimizers of elliptic variational integrals with nearly linear growth, Manuscripta Math. 102 (2000), no. 2, 227-250.
  20. D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, reprint of the 1980 original, Classics in Applied Mathematics, 31, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000.
  21. V. Kokilashvili and M. Krbec, Weighted Inequalities in Lorentz and Orlicz Spaces,World Scientific Publishing Co., Inc., River Edge, NJ, 1991.
  22. N. V. Krylov, Parabolic and elliptic equations with VMO coefficients, Comm. Partial Differential Equations 32 (2007), no. 1-3, 453-475.
  23. N. V. Krylov, Second-order elliptic equations with variably partially VMO coefficients, J. Funct. Anal. 257 (2009), no. 6, 1695-1712.
  24. N. V. Krylov, On parabolic equations in one space dimension, Comm. Partial Differential Equations 41 (2016), no. 4, 644-664.
  25. J. Ok, Gradient continuity for nonlinear obstacle problems, Mediterr. J. Math. 14 (2017), no. 1, Art. 16, 24 pp.
  26. D. K. Palagachev and L. G. Softova, The Calderon-Zygmund property for quasilinear divergence form equations over Reifenberg at domains, Nonlinear Anal. 74 (2011), no. 5, 1721-1730.
  27. N. C. Phuc, Nonlinear Muckenhoupt-Wheeden type bounds on Reifenberg at domains, with applications to quasilinear Riccati type equations, Adv. Math. 250 (2014), 387-419.
  28. J.-F. Rodrigues, Obstacle Problems in Mathematical Physics, North-Holland Mathematics Studies, 134, North-Holland Publishing Co., Amsterdam, 1987.
  29. S. Ryu, Global gradient estimates for nonlinear elliptic equations, J. Korean Math. Soc. 51 (2014), no. 6, 1209-1220.
  30. C. Scheven, Elliptic obstacle problems with measure data: potentials and low order regularity, Publ. Mat. 56 (2012), no. 2, 327-374.
  31. C. Scheven, Gradient potential estimates in non-linear elliptic obstacle problems with measure data, J. Funct. Anal. 262 (2012), no. 6, 2777-2832.
  32. L. G. Softova, Parabolic obstacle problem with measurable coefficients in Morrey-type spaces, in Differential and difference equations with applications, 245-253, Springer Proc. Math. Stat., 164, Springer.
  33. H. Tian and S. Zheng, Lorentz estimates for the gradient of weak solutions to elliptic obstacle problems with partially BMO coefficients, Bound. Value Probl. 2017 (2017), Paper No. 128, 27 pp.
  34. A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Pure and Applied Mathematics, 123, Academic Press, Inc., Orlando, FL, 1986.
  35. T. Toro, Doubling and atness: geometry of measures, Notices Amer. Math. Soc. 44 (1997), no. 9, 1087-1094.
  36. B. O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, Lecture Notes in Mathematics, 1736, Springer-Verlag, Berlin, 2000.


Supported by : National Research Foundation of Korea