Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The regularity and Neumann problem for non-symmetric elliptic operators

Authors: Carlos E. Kenig and David J. Rule
Journal: Trans. Amer. Math. Soc. 361 (2009), 125-160
MSC (2000): Primary 35J25; Secondary 31A25
Published electronically: August 13, 2008
MathSciNet review: 2439401
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We establish optimal $ L^p$ bounds for the non-tangential maximal function of the gradient of the solution to a second-order elliptic operator in divergence form, possibly non-symmetric, with bounded measurable coefficients independent of the vertical variable, on the domain above a Lipschitz graph in the plane, in terms of the $ L^p$-norm at the boundary of the tangential derivative of the Dirichlet data, or of the Neumann data.

References [Enhancements On Off] (What's this?)

  • 1. M. Alfonseca, P. Auscher, A. Axelson, S. Hofmann, and S. Kim, Analyticity of layer potentials and $ L^2$ solvability of boundary value problems for divergence form elliptic equations with complex $ L^\infty$ coefficients, to appear.
  • 2. A.P. Calderón, Cauchy integrals on Lipschitz curves and related operators, Proc. Natl. Acad. Sci. 74 (1977), no. 4, 1324-1327. MR 0466568 (57:6445)
  • 3. R.R. Coifman, A. McIntosh, and Y. Meyer, L'integrale de Cauchy définit un opérateur borné sur $ L^2$ pour les courbes lipschitziennes, Annals of Mathematics 116 (1982), 361-387. MR 672839 (84m:42027)
  • 4. R.R. Coifman, Y. Meyer, and E.M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), no. 2, 304-335. MR 791851 (86i:46029)
  • 5. B.E.J. Dahlberg, Approximation of harmonic functions, Ann. Inst. Fourier 30 (1980), no. 2, 97-107. MR 584274 (82i:31010)
  • 6. -, Weighted norm inequalities for the Lusin area integral and the nontangential maximal functions for functions harmonic in a Lipschitz domain, Studia Mathematica 67 (1980), 297-314. MR 592391 (82f:31003)
  • 7. -, Poisson semigroups and singular integrals, Proc. Amer. Math. Soc. 97 (1986), no. 1, 41-48. MR 831384 (87g:42035)
  • 8. B.E.J. Dahlberg, C.E. Kenig, J. Pipher, and G.C. Verchota, Area integral estimates for higher order elliptic equations and systems, Ann. Inst. Fourier 47 (1997), no. 5, 1425-1461. MR 1600375 (98m:35045)
  • 9. G. David, Opérateurs intégraux singuliers sur certaines courbes du plan complexe, Annales Scientifiques de l'É.N.S. $ 4^{\mbox{\scriptsize e}}$ série 17 (1984), no. 1, 157-189. MR 744071 (85k:42026)
  • 10. G. David, J.-L. Journé, and S. Semmes, Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation, Rev. Mat. Iberoamericana 1 (1985), no. 4, 1-56. MR 850408 (88f:47024)
  • 11. L.C. Evans, Partial differential equations, American Mathematical Society, Providence, Rhode Island, 1999. MR 1625845 (99e:35001)
  • 12. E.B. Fabes, M. Jodeit, Jr., and N.M. Rivière, Potential techniques for boundary value problems on $ C^1$-domains, Acta Math. 141 (1978), 165-187. MR 501367 (80b:31006)
  • 13. D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Heidelberg, 1997.
  • 14. L. Grafakos, Classical and modern Fourier analysis, Pearson Education Inc., Upper Saddle River, New Jersey, 2004.
  • 15. T. Iwaniec and C. Sbordone, Riesz transforms and elliptic PDEs with VMO coefficients, J. Anal. Math. 74 (1998), 183-212. MR 1631658 (99e:35034)
  • 16. D.S. Jerison and C.E. Kenig, The Dirichlet problem in nonsmooth domains, Ann. of Math. 113 (1981), no. 2, 367-382. MR 607897 (84j:35076)
  • 17. J.-L. Journé, Calderón-Zygmund operators, pseudo-differential operators and the Cauchy integral of Calderón, Springer-Verlag, Heidelberg, 1983.
  • 18. C.E. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, CBMS Regional Conference Series, Amer. Math. Soc., Providence, Rhode Island, 1994. MR 1282720 (96a:35040)
  • 19. C.E. Kenig, H. Koch, J. Pipher, and T. Toro, A new approach to absolute continuity of elliptic measure, with applications to non-symmetric equations, Advances in Mathematics 153 (2000), 231-298. MR 1770930 (2002f:35071)
  • 20. C.E. Kenig and W.-M. Ni, On the elliptic equation $ Lu-k+K\,{\rm exp}[2u]=0$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 2, 191-224. MR 829052 (87f:35065)
  • 21. C.E. Kenig and J. Pipher, The Neumann problem for elliptic equations with non-smooth coefficients, Invent. Math. 113 (1993), 447-509. MR 1231834 (95b:35046)
  • 22. J. Pipher, Littlewood-Paley estimates: Some applications to elliptic boundary value problems, CRM Proceedings and Lecture Notes (Providence, Rhode Island), vol. 12, Amer. Math. Soc., 1997, pp. 221-238. MR 1479249 (98j:35047)
  • 23. E.M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, New Jersey, 1970. MR 0290095 (44:7280)
  • 24. -, Harmonic analysis: Real-variable methods, orthogonality and oscillatory integrals, Princeton University Press, Princeton, New Jersey, 1993. MR 1232192 (95c:42002)
  • 25. G.C. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains, Journal of Functional Analysis 59 (1984), 572-611. MR 769382 (86e:35038)
  • 26. G.C. Verchota and A.L. Vogel, Nonsymmetric systems on nonsmooth planar domains, Transactions of the American Mathematical Society 349 (1997), 4501-4535. MR 1443894 (98c:35040)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35J25, 31A25

Retrieve articles in all journals with MSC (2000): 35J25, 31A25

Additional Information

Carlos E. Kenig
Affiliation: Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637

David J. Rule
Affiliation: Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637
Address at time of publication: School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, United Kingdom

Received by editor(s): October 24, 2006
Published electronically: August 13, 2008
Additional Notes: The first author was supported in part by NSF grant number DMS-0456583
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society