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The regularity and Neumann problem for non-symmetric elliptic operators

Author(s): Carlos E. Kenig; David J. Rule
Journal: Trans. Amer. Math. Soc. 361 (2009), 125-160.
MSC (2000): Primary 35J25; Secondary 31A25
Posted: August 13, 2008
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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:

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)

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Additional Information:

Carlos E. Kenig
Affiliation: Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637
Email: cek@math.uchicago.edu

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
Email: rule@uchicago.edu

DOI: 10.1090/S0002-9947-08-04610-2
PII: S 0002-9947(08)04610-2
Received by editor(s): October 24, 2006
Posted: August 13, 2008
Additional Notes: The first author was supported in part by NSF grant number DMS-0456583
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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