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manuscripta mathematica

, Volume 124, Issue 2, pp 139–172 | Cite as

The Green function estimates for strongly elliptic systems of second order

  • Steve HofmannEmail author
  • Seick Kim
Article

Abstract

We establish existence and pointwise estimates of fundamental solutions and Green’s matrices for divergence form, second order strongly elliptic systems in a domain \(\Omega \subseteq {\mathbb{R}}^n, n \geq 3\) , under the assumption that solutions of the system satisfy De Giorgi-Nash type local Hölder continuity estimates. In particular, our results apply to perturbations of diagonal systems, and thus especially to complex perturbations of a single real equation.

Mathematics Subject Classification (2000)

Primary 35A08 35B45 Secondary 35J45 

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References

  1. 1.
    Alfonseca, A., Auscher, P., Axelsson, A., Hofmann, S., Kim, S.: Analyticity of layer potentials and L 2 Solvability of boundary value problems for divergence form elliptic equations with complex \(L^{\infty}\) coefficients, preprintGoogle Scholar
  2. 2.
    Auscher P. (1996). Regularity theorems and heat kernel for elliptic operators. J. London Math. Soc. 54(2): 284–296zbMATHGoogle Scholar
  3. 3.
    Auscher, P., Tchamitchian, Ph.: Square root problem for divergence operators and related topics. Astérisque No. 249 (1998)Google Scholar
  4. 4.
    Campanato S. (1965). Equazioni ellittiche del IIº ordine espazi \({\mathcal{L}}^{(2,\lambda)}\) . (Italian) Ann. Mat. Pura Appl. 69(4): 321–381 zbMATHCrossRefGoogle Scholar
  5. 5.
    De Giorgi E. (1957). Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari. (Italian) Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. 3(3): 25–43Google Scholar
  6. 6.
    Dolzmann G. and Müller S. (1995). Estimates for Green’s matrices of elliptic systems by L p theory. Manuscripta Math. 88(2): 261–273zbMATHCrossRefGoogle Scholar
  7. 7.
    Fuchs M. (1986). The Green matrix for strongly elliptic systems of second order with continuous coefficients. Z. Anal. Anwendungen 5(6): 507–531zbMATHGoogle Scholar
  8. 8.
    Giaquinta M. (1983). Multiple integrals in the calculus of variations and nonlinear elliptic systems. Princeton University Press, PrincetonzbMATHGoogle Scholar
  9. 9.
    Giaquinta M. (1993). Introduction to regularity theory for nonlinear elliptic systems. Birkhäuser Verlag, BaselzbMATHGoogle Scholar
  10. 10.
    Gilbarg D. and Trudinger N.S. (2001). Elliptic partial differential equations of second order. Reprint of the 1998 ed. Springer, BerlinzbMATHGoogle Scholar
  11. 11.
    Grüter M. and Widman K.-O. (1982). The Green function for uniformly elliptic equations. Manuscripta Math. 37(3): 303–342zbMATHCrossRefGoogle Scholar
  12. 12.
    Hofmann S. and Kim S. (2004). Gaussian estimates for fundamental solutions to certain parabolic systems. Publ. Mat. 48: 481–496zbMATHGoogle Scholar
  13. 13.
    John F. and Nirenberg L. (1961). On functions of bounded mean oscillation. Comm. Pure Appl. Math. 14: 415–426zbMATHCrossRefGoogle Scholar
  14. 14.
    Littman W., Stampacchia G. and Weinberger H.F. (1963). Regular points for elliptic equations with discontinuous coefficients. Ann. Scuola Norm. Sup. Pisa 17(3): 43–77zbMATHGoogle Scholar
  15. 15.
    Malý J. and Ziemer W.P. (1997). Fine regularity of solutions of elliptic partial differential equations. American Mathematical Society, ProvidencezbMATHGoogle Scholar
  16. 16.
    Meyers N.G. (1963). An L p-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa 17(3): 189–206zbMATHGoogle Scholar
  17. 17.
    Morrey C.B. Jr. (1966). Multiple integrals in the calculus of variations. Springer, New YorkzbMATHGoogle Scholar
  18. 18.
    Moser J. (1961). On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. 14: 577–591zbMATHCrossRefGoogle Scholar
  19. 19.
    Sarason D. (1975). Functions of vanishing mean oscillation. Trans. Am. Math. Soc. 207: 391–405zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of MissouriColumbiaUSA
  2. 2.Centre for Mathematics and its ApplicationsThe Australian National UniversityCanberraAustralia

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