The mixed problem in Lipschitz domains with general decompositions of the boundary
Authors:
J. L. Taylor, K. A. Ott and R. M. Brown
Journal:
Trans. Amer. Math. Soc. 365 (2013), 2895-2930
MSC (2010):
Primary 35J25, 35J05
DOI:
https://doi.org/10.1090/S0002-9947-2012-05711-4
Published electronically:
December 13, 2012
MathSciNet review:
3034453
Full-text PDF
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Abstract: This paper continues the study of the mixed problem for the Laplacian. We consider a bounded Lipschitz domain ,
, with boundary that is decomposed as
, with
and
disjoint. We let
denote the boundary of
(relative to
) and impose conditions on the dimension and shape of
and the sets
and
. Under these geometric criteria, we show that there exists
depending on the domain
such that for
in the interval
, the mixed problem with Neumann data in the space
and Dirichlet data in the Sobolev space
has a unique solution with the non-tangential maximal function of the gradient of the solution in
. We also obtain results for
when the Dirichlet and Neumann data come from Hardy spaces, and a result when the boundary data comes from weighted Sobolev spaces.
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Additional Information
J. L. Taylor
Affiliation:
Department of Mathematics, Murray State University, Murray, Kentucky 42071-3341
Email:
jtaylor52@murraystate.edu
K. A. Ott
Affiliation:
Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027
Email:
katharine.ott@uky.edu
R. M. Brown
Affiliation:
Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027
Email:
russell.brown@uky.edu
DOI:
https://doi.org/10.1090/S0002-9947-2012-05711-4
Received by editor(s):
May 10, 2011
Published electronically:
December 13, 2012
Additional Notes:
The second author’s research was supported in part by the National Science Foundation.
The third author’s research was supported in part by a grant from the Simons Foundation.
Article copyright:
© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.