Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Nonsymmetric systems
and area integral estimates

Authors: G. C. Verchota and A. L. Vogel
Journal: Proc. Amer. Math. Soc. 128 (2000), 453-462
MSC (1991): Primary 35J55, 31A25
Published electronically: September 27, 1999
MathSciNet review: 1616585
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Even though the $L^2$ Dirichlet problem on Lipschitz domains is not always solvable for nonsymmetric strongly elliptic systems, so that many results and techniques from the symmetric systems are unavailable, there are some similarities with the symmetric systems. We show that the nontangential maximal function and the square function of a solution are equivalent and that there is a Fatou theorem for these solutions.

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

  • [BG72] D. Burkholder and R. Gundy, Distribution function inequalities for the area integral, Studia Math. 44 (1972), 527-544. MR 49:5309
  • [Cal77] A. P. Calderon, Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), 1324-1327. MR 57:6445
  • [CD93] M. Costabel and M. Dauge, Construction of corner singularities for Agmon-Douglis-Nirenberg elliptic systems, Math. Nachr. 162 (1993), 209-237. MR 94k:35090
  • [CMM82] R. R. Coifman, A. McIntosh, and Y. Meyer, L'integrale de Cauchy definit un operateur borne sur $L^2$ pour les courbes Lipschitziennes, Ann. of Math. 116 (1982), 361-387. MR 84m:42027
  • [Dah80] B. E. J. Dahlberg, Weighted norm inequalities for the Lusin area integral and the nontangential maximal function for harmonic functions in Lipschitz domains, Studia Math 65 (1980), 297-314. MR 82f:31003
  • [DK$^{+}$] B. E. J. Dahlberg, C. E. Kenig, J. Pipher, and G. C. Verchota, Area integral estimates and maximum principles for higher order elliptic equations and systems, Ann. Inst. Fourier (Grenoble), 47 (1997), no. 5, 1425-1461. MR 98m:35045
  • [FJR78] E. B. Fabes, M. Jodeit, and N. M. Riviere, Potential techniques for boundary value problems on $C^1$ domains, Acta Math. 141 (1978), 165-186. MR 80b:31006
  • [FS72] C Fefferman and E. Stein, $H^p$ spaces of several variables, Acta Math 129 (1972), 137-193. MR 56:6263
  • [Jou83] Jean-Lin Journé, Calderón-Zygmund Operators, Pseudo-Differential Operators and the Cauchy Integral of calderón, Lecture Notes in Mathematics, vol. 994, Springer-Verlag, Berlin Heidelberg New York Tokyo, 1983. MR 85i:42021
  • [Ken86] C. E. Kenig, Elliptic boundary value problems on Lipschitz domains, Beijing lectures in harmonic analysis, Annals of Math. Studies 112 (1986), 131-183. MR 88a:35066
  • [KKPT98] H. Koch, C. Kenig, J. Pipher, and T. Toro, A new approach to estimates for harmonic measure, and applications to nonsymmetric elliptic equations, preprint.
  • [Koz90] V. A. Kozlov, On the singularities of solutions of the Dirichlet problem for elliptic equations in the neighborhood of corner points, Leningrad Math. J. 1 (1990), 967-982.
  • [KWCQ85] Hua Loo Keng, Lin Wei, and Wu Ci-Quian, Second-order systems of partial differential equations in the plane, Pitman Advanced Publishing Program, Boston London Melbourne, 1985. MR 88m:35002
  • [PV95] J. Pipher and G. C. Verchota, Dilation invariant estimates and the boundary Garding inequality for higher order elliptic operators, Ann. Of Math. 142 (1995), 1-38. MR 96g:35052
  • [Ver84] G. C. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains, J. Funct. Anal. 59 (1984), no. 3, 572-611. MR 86e:35038
  • [VV97] G. C. Verchota and Andrew L. Vogel, Nonsymmetric systems on nonsmooth planar domains, Trans. Amer. Math. Soc. 349 (1997), no. 11, 4501-4535. MR 98c:35040

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 35J55, 31A25

Retrieve articles in all journals with MSC (1991): 35J55, 31A25

Additional Information

G. C. Verchota
Affiliation: Department of Mathematics, Syracuse University, Syracuse, New York 13244

A. L. Vogel
Affiliation: Department of Mathematics, Syracuse University, Syracuse, New York 13244

Keywords: Elliptic, bianalytic, area integral
Received by editor(s): September 16, 1997
Received by editor(s) in revised form: March 16, 1998
Published electronically: September 27, 1999
Additional Notes: The first author was partially supported by NSF Grant DMS-9706648.
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society