Inventiones mathematicae

, Volume 113, Issue 1, pp 447–509

The neumann problem for elliptic equations with non-smooth coefficients

  • Carlos E. Kenig
  • Jill Pipher
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References

  1. [B-A] Beurling, A., Ahlfors, L.: The boundary correspondence under quasiconformal mappings. Acta Math.96, 125–142 (1956)Google Scholar
  2. [C-F-K] Caffarelli, L., Fabes, E., Kenig, C.: Completely singular elliptic-harmonic measures. Indiana Univ. Math. J.30, 917–924 (1981)Google Scholar
  3. [C-F-M-S] Caffarelli, L., Fabes, E., Mortola, S., Salsa, S.: Boundary behavior of nonnegative solutions of elliptic operators in divergence form. Indiana Univ. Math. J.30, 621–640 (1981)Google Scholar
  4. [C-Sc] Calderón, A., Scott, R.: Sobolev type inequalities forp>0. Stud. Math.62, 75–92 (1978)Google Scholar
  5. [C-F] Coifman, R., Fefferman, C.: Weighted norm inequalities for maximal functions and singular integrals. Stud. Math.51, 241–250 (1974)Google Scholar
  6. [D1] Dahlberg, B.: On the Poisson integral for Lipschitz andC 1 domains. Stud. Math.66, 7–24 (1979)Google Scholar
  7. [D2] Dahlberg, B.: On the absolute continuity of elliptic measures. Am. J. Math.108, 1119–1138 (1986)Google Scholar
  8. [D-J-K] Dahlberg, B., Jerison, D., Kening, C.: Area integral estimates for elliptic differential operators with non-smooth coefficients. Ark. Mat.22, 97–108 (1984)Google Scholar
  9. [D-K] Dahlberg, B., Kenig, C.: Hardy spaces and the Neumann problem inL p for Laplace's equation in Lipschitz domains. Ann. Math.125, 437–465 (1987)Google Scholar
  10. [DeG] DeGiorgi, E.: Sulle differenziabilita e analiticita delle estremali degli integrali multipli regulari, Mem. Accad. Sci. Torino3, 25–43 (1957)Google Scholar
  11. [D] Duong, X.T.:H functional calculus of elliptic partial differential operators inL p spaces. Ph.D. Thesis, Macquarie University, Sidney, Australia (1990)Google Scholar
  12. [F-J-K] Fabes, E., Jerison, D., Kenig C.: Necessary and sufficient conditions for absolute continuity of elliptic harmonic measure. Ann. Math.119, 121–141 (1984)Google Scholar
  13. [F-S] Fefferman, C., Stein, E.:H p spaces of several variables. Acta Math.129, 137–192 (1972)Google Scholar
  14. [RF] Fefferman, R.: A criterion for the absolute continuity of the harmonic measure associated with an elliptic operator. J. Am. Math. Soc.2, 127–135 (1989)Google Scholar
  15. [F-K-P] Fefferman, R., Kenig, C., Pipher, J.: The theory of weights and the Dirichlet problem for elliptic equations. Ann. Math.134, 65–124 (1991)Google Scholar
  16. [G] Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Non-linear Elliptic Systems. (Ann. Math. Stud., vol. 105) 1983 Princeton University Press Princeton: NJGoogle Scholar
  17. [G-W] Grüter, M., Widman, K.-O.: The Green function for uniformly elliptic equations. Man. Math.37, 303–342 (1982)Google Scholar
  18. [H-M-W] Hunt, R., Muckenhoupt, B., Wheeden, R.: Weighted norm inequalities for the conjugate function and Hilbert transform. Trans. Am. Math. Soc.176, 227–251 (1973)Google Scholar
  19. [H-W1] Hunt, R., Wheeden, R.: On the boundary values of harmonic functions. Trans. Am. Math. Soc.132, 307–322 (1986)Google Scholar
  20. [H-W2] Hunt, R., Wheeden, R.: Positive harmonic functions on Lipschitz domains. Trans. Am. Math. Soc.147, 507–527 (1970)Google Scholar
  21. [J-K1] Jerison, D., Kenig, C.: The Dirichlet problem in non-smooth domains. Ann. Math.113, 367–382 (1981)Google Scholar
  22. [J-K2] Jerison, D., Kenig, C.: The Neumann problem on Lipschitz domains. Bull. Am. Math. Soc.4, 203–207 (1981)Google Scholar
  23. [J] Jones, P.W.: Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math.147, 71–88 (1981)Google Scholar
  24. [K-P] Kenig, C., Pipher, J.: The oblique derivative on Lipschitz domains withL p data. Am. J. Math.110, 715–737 (1988)Google Scholar
  25. [K-S] Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. (Pure Appl. Math., vol. 88) New York London: Academic Press 1980Google Scholar
  26. [L-S-W] Littman, W., Stampacchia, G., Weinberger, H.: Regular points for elliptic equations with discontinuous coefficients. Ann. Sc. Norm. Super. Pisa17, 43–77 (1983)Google Scholar
  27. [Mo] Moser, J.: On Harnack's theorem for elliptic differential equations, Commun. Pure Appl. Math.14, 577–591 (1961)Google Scholar
  28. [Mu] Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc.165, 207–226 (1972)Google Scholar
  29. [N] Nash, J.: Continuity of the solutions of parabolic and elliptic equations. Am. J. Math.80, 931–954 (1957)Google Scholar
  30. [S-W] Serrin, J., Weinberger, H.: Isolated singularities of solutions of linear elliptic equations. Am. J. Math.88, 158–272 (1966)Google Scholar
  31. [S] Stein, E.: Singular Integrals and Differentiability Properties of Funcitons. Princeton: Princeton University Press 1970Google Scholar
  32. [V] Verchota, G.: Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains. J. Funct. Anal.59, 572–611 (1984)Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Carlos E. Kenig
    • 1
  • Jill Pipher
    • 2
  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Department of MathematicsBrown UniversityProvidenceUSA

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