Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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GRADIENT ESTIMATES FOR ELLIPTIC EQUATIONS IN DIVERGENCE FORM WITH PARTIAL DINI MEAN OSCILLATION COEFFICIENTS

  • Choi, Jongkeun (Department of Mathematics Education Pusan National University) ;
  • Kim, Seick (Department of Mathematics Yonsei University) ;
  • Lee, Kyungrok (Department of Computational Science and Engineering Yonsei University)
  • Received : 2019.11.18
  • Accepted : 2020.03.25
  • Published : 2020.11.01
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Abstract

We provide detailed proofs for local gradient estimates for elliptic equations in divergence form with partial Dini mean oscillation coefficients in a ball and a half ball.

Keywords

Acknowledgement

The authors would like to thank the referee for a very careful reading of the manuscript and many useful comments.

References

  1. A. Ancona, Elliptic operators, conormal derivatives and positive parts of functions, J. Funct. Anal. 257 (2009), no. 7, 2124-2158. https://doi.org/10.1016/j.jfa.2008.12.019
  2. H. Brezis, On a conjecture of J. Serrin, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 19 (2008), no. 4, 335-338. https://doi.org/10.4171/RLM/529
  3. S.-S. Byun and L. Wang, Elliptic equations with measurable coefficients in Reifenberg domains, Adv. Math. 225 (2010), no. 5, 2648-2673. https://doi.org/10.1016/j.aim.2010.05.014
  4. J. Choi and H. Dong, Gradient estimates for Stokes systems in domains, Dyn. Partial Differ. Equ. 16 (2019), no. 1, 1-24. https://doi.org/10.4310/dpde.2019.v16.n1.a1
  5. J. Choi and H. Dong, Gradient estimates for Stokes systems with Dini mean oscillation coefficients, J. Differential Equations 266 (2019), no. 8, 4451-4509. https://doi.org/10.1016/j.jde.2018.10.001
  6. J. Choi, H. Dong, and D. Kim, Conormal derivative problems for stationary Stokes system in Sobolev spaces, Discrete Contin. Dyn. Syst. 38 (2018), no. 5, 2349-2374. https://doi.org/10.3934/dcds.2018097
  7. H. Dong, Gradient estimates for parabolic and elliptic systems from linear laminates, Arch. Ration. Mech. Anal. 205 (2012), no. 1, 119-149. https://doi.org/10.1007/s00205-012-0501-z
  8. H. Dong, L. Escauriaza, and S. Kim, On $C^1$, $C^2$, and weak type-(1, 1) estimates for linear elliptic operators: part II, Math. Ann. 370 (2018), no. 1-2, 447-489. https://doi.org/10.1007/s00208-017-1603-6
  9. 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
  10. 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. https://doi.org/10.1016/j.jfa.2011.08.001
  11. H. Dong and D. Kim, Parabolic and elliptic systems in divergence form with variably partially BMO coefficients, SIAM J. Math. Anal. 43 (2011), no. 3, 1075-1098. https://doi.org/10.1137/100794614
  12. H. Dong and S. Kim, Partial Schauder estimates for second-order elliptic and parabolic equations, Calc. Var. Partial Differential Equations 40 (2011), no. 3-4, 481-500. https://doi.org/10.1007/s00526-010-0348-9
  13. H. Dong and S. Kim, On $C^1$, $C^2$, and weak type-(1, 1) estimates for linear elliptic operators, Comm. Partial Differential Equations 42 (2017), no. 3, 417-435. https://doi.org/10.1080/03605302.2017.1278773
  14. H. Dong and S. Kim, Partial Schauder estimates for second-order elliptic and parabolic equations: a revisit, Int. Math. Res. Not. IMRN 2019 (2019), no. 7, 2085-2136. https://doi.org/10.1093/imrn/rnx180
  15. H. Dong, J. Lee, and S. Kim, On conormal and oblique derivative problem for elliptic equations with Dini mean oscillation coefficients, arXiv:1801.09836.
  16. H. Dong and Z. Li, $C^2$ estimate for oblique derivative problem with mean Dini coefficients, arXiv:1904.02766.
  17. H. Dong and J. Xiong, Boundary gradient estimates for parabolic and elliptic systems from linear laminates, Int. Math. Res. Not. IMRN 2015 (2015), no. 17, 7734-7756. https://doi.org/10.1093/imrn/rnu185
  18. H. Dong and L. Xu, Hessian estimates for non-divergence form elliptic equations arising from composite materials, arXiv:1904.10950.
  19. H. Dong and L. Xu, Gradient estimates for divergence form elliptic systems arising from composite material, SIAM J. Math. Anal. 51 (2019), no. 3, 2444-2478. https://doi.org/10.1137/18M1226658
  20. L. Escauriaza and S. Montaner, Some remarks on the $L^p$ regularity of second derivatives of solutions to non-divergence elliptic equations and the Dini condition, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28 (2017), no. 1, 49-63. https://doi.org/10.4171/RLM/751
  21. P. Fife, Schauder estimates under incomplete Holder continuity assumptions, Pacific J. Math. 13 (1963), 511-550. https://doi.org/10.2140/pjm.1963.13.511
  22. M. Giaquinta, Introduction to regularity theory for nonlinear elliptic systems, Lectures in Mathematics ETH Zurich, Birkhauser Verlag, Basel, 1993.
  23. Y. Jin, D. Li, and X.-J. Wang, Regularity and analyticity of solutions in a direction for elliptic equations, Pacific J. Math. 276 (2015), no. 2, 419-436. https://doi.org/10.2140/pjm.2015.276.419
  24. G. Tian and X.-J. Wang, Partial regularity for elliptic equations, Discrete Contin. Dyn. Syst. 28 (2010), no. 3, 899-913. https://doi.org/10.3934/dcds.2010.28.899