A note on the Dirichlet problem for the Stokes system in Lipschitz domains
Author:
Zhong Wei Shen
Journal:
Proc. Amer. Math. Soc. 123 (1995), 801811
MSC:
Primary 35Q30; Secondary 76D07
MathSciNet review:
1223521
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Abstract: We study the Dirichlet problem for the Stokes system in Lipschitz domains. Optimal estimates are obtained when the dimension . In the case of , we establish a weak estimate of solutions for certain range of p.
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 , estimates for the threedimension system of elastostatics on Lipschitz domains, Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, 1990.
 [DKPV]
 B. Dahlberg, C. Kenig, J. Pipher, and G. Verchota, Area integral estimates and maximum principles for higher order elliptic equations and systems on Lipschitz domains, preprint.
 [DKV]
 B. Dahlberg, C. Kenig, and G. Verchota, Boundary value problems for the systems of elastostatics in Lipschitz domains, Duke Math. J. 57 (1988), 795818. MR 975122 (90d:35259)
 [F]
 E. Fabes, Layer potential methods for boundary value problems on Lipschitz domains, Lecture Notes in Math., vol. 1344, SpringerVerlag, New York, 1987, pp. 5580. MR 973881
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 E. Fabes, C. Kenig, and G. Verchota, The Dirichlet problem for the Stokes system on Lipschitz domains, Duke Math. J. 57 (1988), 769793. MR 975121 (90d:35258)
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 M. Giaquinta and G. Modica, Nonlinear systems of the type of the stationary NavierStokes system, J. Reine Angew. Math. 330 (1982), 173214. MR 641818 (83h:35041)
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 D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, preprint. MR 1331981 (96b:35042)
 [MP]
 V. G. Maz'ya and V. A. Plamenvskii, On properties of solutions of threedimensional problems of elasticity theory and hydrodynamics in domains with isolated singular points, Amer. Math. Soc. Transl. Ser. 2, vol. 123, Amer. Math. Soc., Providence, RI, 1984, pp. 109123.
 [PV]
 J. Pipher and G. Verchota, The Dirichlet problem in for biharmonic functions on Lipschitz domains, Amer. J. Math. 114 (1992), 923972. MR 1183527 (94g:35069)
 [S]
 E. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, NJ, 1970. MR 0290095 (44:7280)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199512235219
PII:
S 00029939(1995)12235219
Article copyright:
© Copyright 1995
American Mathematical Society
