Available in electronic format
Available in print format		 Transacrions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Potential theory on Lipschitz domains in Riemannian manifolds: The case of Dini metric tensors

Author(s): Marius Mitrea; Michael Taylor
Journal: Trans. Amer. Math. Soc. 355 (2003), 1961-1985.
MSC (2000): Primary 31C12, 35J25, 45E05
Posted: November 14, 2002
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

Abstract: We study the applicability of the method of layer potentials in the treatment of boundary value problems for the Laplace-Beltrami operator on Lipschitz sub-domains of Riemannian manifolds, in the case when the metric tensor $g_{jk} dx_j\otimes dx_k$ has low regularity. Under the assumption that

\begin{displaymath}\vert g_{jk}(x)-g_{jk}(y)\vert\leq C\,\omega(\vert x-y\vert),\end{displaymath}

where the modulus of continuity $\omega$ satisfies a Dini-type condition, we prove the well-posedness of the classical Dirichlet and Neumann problems with $L^p$ boundary data, for sharp ranges of $p$'s and with optimal nontangential maximal function estimates.

References:

[Ca]
A.P. Calderón, Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. (USA) 74 (1977), 1324-1327. MR 57:6445
[CMM]
R. Coifman, A. McIntosh, and Y. Meyer, L'intégrale de Cauchy definit un opérateur borné sur $L^2$ pour les courbes lipschitziennes, Ann. Math. 116 (1982), 361-388. MR 84m:42027
[DK]
B. Dahlberg and C. Kenig, Hardy spaces and the Neumann problem in $L^p$ for Laplace's equation in Lipschitz domains, Ann. of Math. 125 (1987), 437-465. MR 88d:35044
[DiF]
G. DiFazio, $L^p$ estimates for divergence form elliptic equations with discontinuous coefficients, Boll. U.M.I. 10-A (1996), 409-420. MR 97e:35034
[Ed]
R. E. Edwards, Functional Analysis. Theory and Applications, Corrected reprint of the 1965 original. Dover Publications, Inc., New York, 1995. MR 95k:46001
[FJR]
E. Fabes, M. Jodeit, and N. Riviere, Potential techniques for boundary value problems in $C^1$ domains, Acta Math. 114 (1978), 165-186. MR 80b:31006
[GW]
M. Grüter and K.-O. Widman, The Green function for uniformly elliptic equations, Manuscr. Math. 37 (1982), 303-342. MR 83h:35033
[JK]
D. Jerison and C. Kenig, Boundary value problems in Lipschitz domains, pp. 1-68 in ``Studies in Partial Differential Equations,'' W. Littman, ed., MAA, 1982. MR 85f:35057
[KM]
N. Kalton and M. Mitrea, Stability results on interpolation scales of quasi-Banach spaces and applications, Trans. Amer. Math. Soc. 350 (1998), 3903-3922. MR 98m:46094
[KP]
C. Kenig and J. Pipher, The Neumann problem for elliptic equations with non-smooth coefficients, Invent. Math. 113 (1993), 447-509. MR 95b:35046
[Me]
N.G.Meyers, An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations, Annali della Scuola Norm. Sup. Pisa, III, 17 (1963), 189-206. MR 28:2328
[MT1]
M. Mitrea and M. Taylor, Boundary layer methods for Lipschitz domains in Riemannian manifolds, J. Funct. Anal. 163 (1999), 181-251. MR 2000b:35050
[MT2]
M. Mitrea and M. Taylor, Potential theory on Lipschitz domains in Riemannian manifolds: $L^p$, Hardy, and Hölder space results, Communications in Analysis and Geometry, 9 (2001), 369-421. MR 2002f:31012
[MT3]
M. Mitrea and M. Taylor, Potential theory on Lipschitz domains in Riemannian manifolds: Sobolev-Besov space results and the Poisson problem, J. Funct. Anal. 176 (2000), 1-79. MR 2002e:58044
[MT4]
M. Mitrea and M. Taylor, Potential theory on Lipschitz domains in Riemannian manifolds: Hölder continuous metric tensors, Comm. PDE 25 (2000), 1487-1536. MR 2001h:35040
[P]
J. Peetre, On convolution operators leaving $L^{p\lambda}$-spaces invariant, Ann. Mat. Pura Appl. 72 (1966), 295-304. MR 35:812
[Sp]
S. Spanne, Some function spaces defined using the mean oscillation over cubes, Ann. Scuola Norm. Sup. Pisa 19 (1965), 593-608. MR 32:8140
[T1]
M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, Boston, 1991. MR 92j:35193
[T2]
M. Taylor, Microlocal analysis on Morrey spaces, in Singularities and Oscillations (J. Rauch and M. Taylor, eds.), IMA Vol. 91, Springer-Verlag, New York, 1997, pp. 97-135. MR 99a:35010
[T3]
M. Taylor, Tools for PDE: Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Mathematical Surveys and Monographs, 81, Amer. Math. Soc., Providence, RI, 2000. MR 2001g:35004
[Ve]
G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains, J. Funct. Anal. 59 (1984), 572-611. MR 86e:35038

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 31C12, 35J25, 45E05

Retrieve articles in all Journals with MSC (2000): 31C12, 35J25, 45E05

Additional Information:

Marius Mitrea
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: marius@math.missouri.edu

Michael Taylor
Affiliation: Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599
Email: met@math.unc.edu

DOI: 10.1090/S0002-9947-02-03150-1
PII: S 0002-9947(02)03150-1
Received by editor(s): April 24, 2002
Received by editor(s) in revised form: July 4, 2002
Posted: November 14, 2002
Additional Notes: The first author was partially supported by NSF grants DMS-9870018 and DMS-0139801
The second author was partially supported by NSF grant DMS-9877077
Copyright of article: Copyright 2002, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google