×

The Dirichlet problem for the Stokes system on Lipschitz domains. (English) Zbl 0685.35085

The purpose of this paper is to study the solvability of the Dirichlet problem for the Stokes system of linearized hydrostatics on an arbitrary domain. The authors also present existence and uniqueness with optimal error estimate of the developed method. The authors’ result is proven by establishing the relevant Rellich formulas for Stokes operators.
Reviewer: P.K.Mahanti

MSC:

35Q30 Navier-Stokes equations
35A35 Theoretical approximation in context of PDEs
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] A. P. Calderón, Cauchy integrals on Lipschitz curves and related operators , Proc. Nat. Acad. Sc. U.S.A. 74 (1977), no. 4, 1324-1327. JSTOR: · Zbl 0373.44003
[2] A. P. Calderón, Boundary value problems for the Laplace equation in Lipschitzian domains , Recent progress in Fourier analysis (El Escorial, 1983), North Holland Mathematics Studies, vol. 111, North-Holland, Amsterdam, 1985, pp. 33-48. · Zbl 0608.31001
[3] R. R. Coifman, A. McIntosh, and Y. Meyer, L’intégrale de Cauchy définit un opérateur borné sur \(L^2\) pour les courbes lipschitziennes , Ann. of Math. (2) 116 (1982), no. 2, 361-387. JSTOR: · Zbl 0497.42012
[4] B. E. J. Dahlberg, Estimates of harmonic measure , Arch. Rational Mech. Anal. 65 (1977), no. 3, 275-288. · Zbl 0406.28009
[5] B. E. J. Dahlberg, On the Poisson integral for Lipschitz and \(C^1\)-domains , Studia Math. 66 (1979), no. 1, 13-24. · Zbl 0422.31008
[6] B. E. J. Dahlberg and C. E. Kenig, Hardy spaces and the Neumann problem in \(L^ p\) for Laplace’s equation in Lipschitz domains , Ann. of Math. (2) 125 (1987), no. 3, 437-465. JSTOR: · Zbl 0658.35027
[7] B. E. J. Dahlberg, C. E. Kenig, and G. C. Verchota, The Dirichlet problem for the biharmonic equation in a Lipschitz domain , Ann. Inst. Fourier (Grenoble) 36 (1986), no. 3, 109-135. · Zbl 0589.35040
[8] B. E. J. Dahlberg, C. E. Kenig, and G. C. Verchota, Boundary value problems for the systems of elastostatics in Lipschitz domains , Duke Math. J. 57 (1988), no. 3, 795-818. · Zbl 0699.35073
[9] E. B. Fabes, Boundary value problems of linear elastostatics and hydrostatics on Lipschitz domains , Miniconference on Linear Analysis and Function Spaces (Canberra, 1984), Proceedings of the Centre Mathematical Analysis, A.N.U., vol. 9, Austral. Nat. Univ., Canberra, 1985, pp. 27-45. · Zbl 0608.35014
[10] E. B. Fabes, Max Jodeit, Jr., and Jeff E. Lewis, Double layer potentials for domains with corners and edges , Indiana Univ. Math. J. 26 (1977), no. 1, 95-114. · Zbl 0363.35010
[11] E. B. Fabes, M. Jodeit, Jr., and N. M. Rivière, Potential techniques for boundary value problems on \(C^1\)-domains , Acta Math. 141 (1978), no. 3-4, 165-186. · Zbl 0402.31009
[12] D. S. Jerison and C. E. Kenig, The Dirichlet problem in nonsmooth domains , Ann. of Math. (2) 113 (1981), no. 2, 367-382. JSTOR: · Zbl 0434.35027
[13] D. S. Jerison and C. E. Kenig, The Neumann problem on Lipschitz domains , Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 203-207. · Zbl 0471.35026
[14] C. E. Kenig, Boundary value problems of linear elastostatics and hydrostatics on Lipschitz domains , Seminaire Goulaouic-Meyer-Schwartz, 1983-1984, XXI, École Polytechnique, Palaiseau, France, 1984. · Zbl 0547.73007
[15] C. E. Kenig, Recent progress on boundary value problems on Lipschitz domains , Pseudodifferential Operators and Applications (Notre Dame, Ind., 1984), Proc. Sympos. Pure Math., vol. 43, Amer. Math. Soc., Providence, RI, 1985, pp. 175-205. · Zbl 0593.35038
[16] C. E. Kenig, Elliptic boundary value problems on Lipschitz domains , Beijing Lectures in Harmonic Analysis (Beijing, 1984), Annals of Math. Studies, vol. 112, Princeton Univ. Press, Princeton, NJ, 1986, pp. 131-183. · Zbl 0624.35029
[17] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow , Revised English edition. Translated from the Russian by Richard A. Silverman, Gordon and Breach, New York, 1963. · Zbl 0121.42701
[18] J. Nečas, Les méthodes directes en théorie des équations élliptiques , Academia, Prague, 1967. · Zbl 1225.35003
[19] F. Rellich, Darstellung der Eigenwerte von \(\Delta u+\lambda u=0\) durch ein Randintegral , Math. Z. 46 (1940), 635-636. · Zbl 0023.04204
[20] 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. · Zbl 0589.31005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.