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Global Estimates for Green’s Matrix of Second Order Parabolic Systems with Application to Elliptic Systems in Two Dimensional Domains

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Abstract

We establish global Gaussian estimates for the Green’s matrix of divergence form, second order parabolic systems in a cylindrical domain under the assumption that weak solutions of the system vanishing on a portion of the boundary satisfy a certain local boundedness estimate and a local Hölder estimate. From these estimates, we also derive global estimates for the Green’s matrix for elliptic systems with bounded measurable coefficients in two dimensional domains. We present a unified approach valid for both the scalar and vectorial cases and discuss several applications of our result.

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References

  1. Aronson, D.G.: Bounds for the fundamental solution of a parabolic equation. Bull. Am. Math. Soc. 73, 890–896 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  2. Auscher, P.: Regularity theorems and heat kernel for elliptic operators. J. Lond. Math. Soc. (2) 54(2), 284–296 (1996)

    MATH  MathSciNet  Google Scholar 

  3. Auscher, P., McIntosh, A., Tchamitchian, P.h.: Heat kernels of second order complex elliptic operators and applications. J. Funct. Anal. 152(1), 22–73 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Auscher, P., Qafsaoui, M.: Equivalence between regularity theorems and heat kernel estimates for higher order elliptic operators and systems under divergence form. J. Funct. Anal. 177(2), 310–364 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Auscher, P., Tchamitchian, P.h.: Square Root Problem for Divergence Operators and Related Topics. Astérisque No. 249 (1998)

  6. Auscher, P., Tchamitchian, P.h.: Gaussian estimates for second order elliptic divergence operators on Lipschitz and \(C\sp 1\) domains. In: Evolution Equations and their Applications in Physical and Life Sciences (Bad Herrenalb, 1998). Lecture Notes in Pure and Appl. Math., vol. 215, pp. 15–32. Dekker, New York (2001)

    Google Scholar 

  7. Cho, S.: Two-sided global estimates of the Green’s function of parabolic equations. Potential Anal. 25(4), 387–398 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cho, S., Dong, H., Kim, S.: On the Green’s matrices of strongly parabolic systems of second order. Indiana Univ. Math. J. 57(4), 1633–1677 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Davies, E.B.: Explicit constants for Gaussian upper bounds on heat kernels. Am. J. Math. 109(2), 319–333 (1987)

    Article  MATH  Google Scholar 

  10. Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge Univ. Press, Cambridge, UK (1989)

    Book  MATH  Google Scholar 

  11. Dong, H., Kim, D.: On the L p -solvability of higher order parabolic and elliptic systems with BMO coefficients. Arch. Rational Mech. Anal. 199(3), 889–941 (2011). doi:10.1007/s00205-010-0345-3

  12. Dong, H., Kim, S.: Green’s matrices of second order elliptic systems with measurable coefficients in two dimensional domains. Trans. Am. Math. Soc. 361(6), 3303–3323 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Dolzmann, G., Müller, S.: Estimates for Green’s matrices of elliptic systems by \(L\sp p\) theory. Manuscr. Math. 88(2), 261–273 (1995)

    Article  MATH  Google Scholar 

  14. Eidel’man, S.D.: Parabolic Systems. Translated from the Russian by Scripta Technica. London North-Holland Publishing Co., Amsterdam-London, Wolters-Noordhoff Publishing, Groningen (1969)

    MATH  Google Scholar 

  15. Fabes, E.B., Stroock, D.W.: A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash. Arch. Ration. Mech. Anal. 96(4), 327–338 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  16. Fuchs, M.: The Green-matrix for elliptic systems which satisfy the Legendre-Hadamard condition. Manuscr. Math. 46(1–3), 97–115 (1984)

    Article  MATH  Google Scholar 

  17. Fuchs, M.: The Green matrix for strongly elliptic systems of second order with continuous coefficients. Z. Anal. Ihre Anwend. 5(6), 507–531 (1986)

    MATH  Google Scholar 

  18. Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton University Press, Princeton, NJ (1983)

    MATH  Google Scholar 

  19. Giaquinta, M.: Introduction to Regularity Theory for Nonlinear Elliptic Systems. Birkhäuser Verlag, Basel (1993)

    MATH  Google Scholar 

  20. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Reprint of the 1998 edn. Springer, Berlin (2001)

    MATH  Google Scholar 

  21. Grunau, H.-C., Robert, F.: Positivity and almost positivity of biharmonic Green’s functions under Dirichlet boundary conditions. Arch. Ration. Mech. Anal. 195(3), 865–898 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  22. Grunau, H.-C., Sweers, G.: Regions of positivity for polyharmonic Green functions in arbitrary domains. Proc. Am. Math. Soc. 135(11), 3537–3546 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  23. Grüter, M., Widman, K.-O.: The Green function for uniformly elliptic equations. Manuscr. Math. 37(3), 303–342 (1982)

    Article  MATH  Google Scholar 

  24. Hofmann, S., Kim, S.: Gaussian estimates for fundamental solutions to certain parabolic systems. Publ. Mat. 48, 481–496 (2004)

    MATH  MathSciNet  Google Scholar 

  25. Hofmann, S., Kim, S.: The Green function estimates for strongly elliptic systems of second order. Manuscr. Math. 124(2), 139–172 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  26. Kang, K., Kim, S.: Global pointwise estimates for Green’s matrix of second order elliptic systems. J. Diff. Equ. 249(11), 2643–2662 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  27. Kim, S.: Gaussian estimates for fundamental solutions of second order parabolic systems with time-independent coefficients. Trans. Am. Math. Soc. 360(11), 6031–6043 (2008)

    Article  MATH  Google Scholar 

  28. Krylov N.V.: Parabolic and elliptic equations with VMO coefficients. Commun. Partial Differ. Equ. 32(3), 453–475 (2007)

    Article  MATH  Google Scholar 

  29. Kukavica, I., Nyström, K.: Unique continuation on the boundary for Dini domains. Proc. Am. Math. Soc. 126(2), 441–446 (1998)

    Article  MATH  Google Scholar 

  30. Ladyženskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, Providence, RI (1967)

    Google Scholar 

  31. Lieberman, G.M.: Hölder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions. Ann. Mat. Pura Appl. 148(4), 77–99 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  32. Lieberman G.M.: Second Order Parabolic Differential Equations. World Scientific Publishing Co., Inc., River Edge, NJ (1996)

    MATH  Google Scholar 

  33. Littman, W., Stampacchia, G., Weinberger, H.F.: Regular points for elliptic equations with discontinuous coefficients. Ann. Sc. Norm. Super. Pisa 17(3), 43–77 (1963)

    MATH  MathSciNet  Google Scholar 

  34. Malý, J., Ziemer, W.P.: Fine Regularity of Solutions of Elliptic Partial Differential Equations. American Mathematical Society, Providence, RI (1997)

    MATH  Google Scholar 

  35. Moser, J.: A Harnack inequality for parabolic differential equations. Commun. Pure Appl. Math. 17, 101–134 (1964)

    Article  MATH  Google Scholar 

  36. Nash, J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  37. Porper, F.O., Eidel’man, S.D.: Two-sided estimates of the fundamental solutions of second-order parabolic equations and some applications of them. Usp. Mat. Nauk 39(3)(237), 107–156 (1984, in Russian); English translation: Russ. Math. Surv. 39(3), 119–179 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  38. Robinson, D.W.: Elliptic Operators and Lie Groups. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1991)

    Google Scholar 

  39. Struwe, M.: On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems. Manuscr. Math. 35(1–2), 125–145 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  40. Varopoulos, N.Th., Saloff-Coste, L., Coulhon, T.: Analysis and Geometry on Groups. Cambridge Tracts in Mathematics, vol. 100. Cambridge University Press, Cambridge (1992)

    Google Scholar 

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Author notes

  1. Sungwon Cho

    Present address: Department of Mathematics Education, Gwangju National University of Education, Gwangju, 500-703, Republic of Korea

  2. Seick Kim

    Present address: Department of Computational Science and Engineering, Yonsei University, Seoul, 120-749, Republic of Korea

Authors and Affiliations

  1. Department of Mathematics, Yonsei University, Seoul, 120-749, Republic of Korea

    Sungwon Cho & Seick Kim

  2. Division of Applied Mathematics, Brown University, 182 George Street, Box F, Providence, RI, 02912, USA

    Hongjie Dong

Authors
  1. Sungwon Cho
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  2. Hongjie Dong
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  3. Seick Kim
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Correspondence to Seick Kim.

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Cho, S., Dong, H. & Kim, S. Global Estimates for Green’s Matrix of Second Order Parabolic Systems with Application to Elliptic Systems in Two Dimensional Domains. Potential Anal 36, 339–372 (2012). https://doi.org/10.1007/s11118-011-9234-0

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  • DOI: https://doi.org/10.1007/s11118-011-9234-0

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