Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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GLOBAL REGULARITY OF SOLUTIONS TO QUASILINEAR CONORMAL DERIVATIVE PROBLEM WITH CONTROLLED GROWTH

  • Kim, Do-Yoon (Department of Applied Mathematics Kyung Hee University)
  • Received : 2011.08.09
  • Published : 2012.11.01
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Abstract

We prove the global regularity of weak solutions to a conormal derivative boundary value problem for quasilinear elliptic equations in divergence form on Lipschitz domains under the controlled growth conditions on the low order terms. The leading coefficients are in the class of BMO functions with small mean oscillations.

Keywords

Acknowledgement

Supported by : Kyung Hee University

References

  1. A. A. Arkhipova, Partial regularity of solutions of quasilinear elliptic systems with a nonsmooth condition for a conormal derivative, Mat. Sb. 184 (1993), no. 2, 87-104; translation in Russian Acad. Sci. Sb. Math. 78 (1994), no. 1, 215-230.
  2. A. A. Arkhipova, On the regularity of the solution of the Neumann problem for quasilinear parabolic systems, Izv. Ross. Akad. Nauk Ser. Mat. 58 (1994), no. 5, 3-25; translation in Russian Acad. Sci. Izv. Math. 45 (1995), no. 2, 231-253.
  3. A. A. Arkhipova, Reverse Holder inequalities with boundary integrals and $L_{p}$-estimates for solutions of nonlinear elliptic and parabolic boundary-value problems, Nonlinear evolution equations, 15-42, Amer. Math. Soc. Transl. Ser. 2, 164, Amer. Math. Soc., Providence, RI, 1995.
  4. H. Dong and D. Kim, Elliptic equations in divergence form with partially BMO coefficients, Arch. Ration. Mech. Anal. 196 (2010), no. 1, 25-70. https://doi.org/10.1007/s00205-009-0228-7
  5. 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. https://doi.org/10.1007/s00526-010-0344-0
  6. H. Dong and D. Kim, Global regularity of weak solutions to quasilinear elliptic and parabolic equations with controlled growth, Comm. Partial Differential Equations 36 (2011), no. 10, 1750- 1777. https://doi.org/10.1080/03605302.2011.571746
  7. M. Giaquinta, A counter-example to the boundary regularity of solutions to elliptic quasilinear systems, Manuscripta Math. 24 (1978), no. 2, 217-220. https://doi.org/10.1007/BF01310055
  8. M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, Princeton, NJ, 1983.
  9. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001.
  10. O. A. Ladyhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis Academic Press, New York-London 1968.
  11. O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society: Providence, RI, 1967.
  12. G. M. Lieberman, The conormal derivative problem for elliptic equations of variational type, J. Differential Equations 49 (1983), no. 2, 218-257. https://doi.org/10.1016/0022-0396(83)90013-X
  13. D. K. Palagachev, Global Holder continuity of weak solutions to quasilinear divergence form elliptic equations, J. Math. Anal. Appl. 359 (2009), no. 1, 159-167. https://doi.org/10.1016/j.jmaa.2009.05.044
  14. D. K. Palagachev, Quasilinear divergence form elliptic equations in rough domains, Complex Var. Elliptic Equ. 55 (2010), no. 5-6, 581-591. https://doi.org/10.1080/17476930903276159
  15. D. K. Palagachev and L. G. Softova, The Calderon-Zygmund property for quasilinear divergence form equations over Reifenberg flat domains, Nonlinear Anal. 74 (2011), no. 5, 1721-1730. https://doi.org/10.1016/j.na.2010.10.044
  16. J. Stara, O. John, and J. Maly, Counterexamples to the regularity of weak solutions of the quasilinear parabolic system, Comment. Math. Univ. Carolin. 27 (1986), no. 1, 123-136.
  17. P. Winkert, $L^{\infty}$-estimates for nonlinear elliptic Neumann boundary value problems, Nonlinear Differential Equations Appl. 17 (2010), no. 3, 289-302. https://doi.org/10.1007/s00030-009-0054-5
  18. P. Winkert and R. Zacher, A priori bounds for weak solutions to elliptic equations with nonstandard growth, Discrete Contin. Dyn. Syst. Ser. S 5 (2012), no. 4, 865-878.

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