Rocky Mountain Journal of Mathematics

Optimal Morrey estimate for parabolic equations in divergence form via Green's functions

Junjie Zhang and Shenzhou Zheng

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

This paper presents a local Morrey regularity with the optimal exponents for linear parabolic equations in divergence form under the assumption that the leading coefficient is independent of $t$ and not necessarily symmetric based on a rather different approach. Here, we achieve it by applying natural growth properties of Green's functions through the use of parabolic operators and the hole-filling technique.

Article information

Source
Rocky Mountain J. Math., Volume 48, Number 7 (2018), 2431-2457.

Dates
First available in Project Euclid: 14 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1544756816

Digital Object Identifier
doi:10.1216/RMJ-2018-48-7-2431

Mathematical Reviews number (MathSciNet)
MR3892139

Zentralblatt MATH identifier
06999269

Subjects
Primary: 35D30: Weak solutions 35K10: Second-order parabolic equations

Keywords
Green's function Morrey regularity parabolic equations Gaussian estimates hole-filling technique

Citation

Zhang, Junjie; Zheng, Shenzhou. Optimal Morrey estimate for parabolic equations in divergence form via Green's functions. Rocky Mountain J. Math. 48 (2018), no. 7, 2431--2457. doi:10.1216/RMJ-2018-48-7-2431. https://projecteuclid.org/euclid.rmjm/1544756816


Export citation

References

  • G. Alexander and T. Andras, Two-sided estimates of heat kernels on metric measure spaces, Ann. Probab. 40 (2012), 1212–1284.
  • D.G. Aronson, Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc. 73 (1967), 890–896.
  • A. Bensoussan and J. Frehse, Regularity results for nonlinear elliptic systems and applications, Spinger-Verlag, Berlin, 2002.
  • S. Cho, Two-sided global estimates of the Green's function of parabolic equations, Potential Anal. 25 (2006), 387–398.
  • S. Cho, H.J. Dong and S. Kim, Global estimates for Green's matrix of second order parabolic systems with application to elliptic systems in two dimensional domains, Potential Anal. 36 (2012), 339–372.
  • ––––, On the Green's matrices of strongly parabolic systems of second order, Indiana Univ. Math. J. 57 (2008), 1633–1677.
  • J. Choi and S. Kim, Neumann function for second order elliptic systems with measurable coefficients, Trans. Amer. Math. Soc. 365 (2013), 6283–6307.
  • ––––, Green's function for second order parabolic systems with Neumann boundary condition, J. Differ. Eqs. 254 (2013), 2834–2860.
  • E.B. Davies, The equivalence of certain heat kernel and Green function bounds, J. Funct. Anal. 71 (1987), 88–103.
  • ––––, Heat kernel and spectral theory, Cambridge University Press, Cambridge, 1989.
  • E. Dibenedetto and J. Manfredi, On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems, Amer. J. Math. 115 (1993), 1107–1134.
  • H.J. Dong and S. Kim, Green's functions for parabolic systems of second order in time-varying domains, Comm. Pure Appl. Anal. 13 (2014), 1407–1433.
  • Z.S. Feng, S.Z. Zheng and H.F. Lu, Green's function of non-linear degenerate elliptic operators and its application to regularity, Differ. Integ. Eqs. 21 (2008), 717–741.
  • D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin, 2001.
  • C.E. Grüter and K.O. Widman, The Green function for uniformly elliptic equations, Manuscr. Math. 37 (1982), 303–342.
  • S. Hofmann and S. Kim, Gaussian estimates for fundamental solutions to certain parabolic systems, Publ. Mat. 48 (2004), 481–496.
  • T. Huang and C.Y. Wang, Notes on the regularity of harmonic map systems, Proc. Amer. Math. Soc. 138 (2010), 2015–2023.
  • K. Kang and S. Kim, Global pointwise estimates for Green's matrix of second order elliptic systems, J. Differ. Eqs. 249 (2010), 2643–2662.
  • S. Kim, Gaussian estimates for fundamental solutions of second order parabolic systems with time-independent coefficients, Trans. Amer. Math. Soc. 360 (2008), 6031–6043.
  • W. Littman, G. Stampacchia and H.F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuol. Norm. Sup. Pisa 17 (1963), 43–77.
  • G. Mazzoni, Green function for X-elliptic operators, Manuscr. Math. 115 (2004), 207–238.
  • J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931–954.
  • M. Struwe, On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems, Manuscr. Math. 35 (1981), 125–145.
  • J.L. Taylor, S. Kim and R.M. Brown, The Green function for elliptic systems in two dimensions, Comm. Part. Differ. Eqs. 38 (2013), 1574–1600.
  • S.Z. Zheng and X.Y. Kang, The comparison of Green function for quasilinear elliptic equations, Acta Math. Sci. 25 (2005), 470–480.