Abstract and Applied Analysis

On the Dirichlet Problem for the Stokes System in Multiply Connected Domains

Alberto Cialdea, Vita Leonessa, and Angelica Malaspina

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Abstract

The Dirichlet problem for the Stokes system in a multiply connected domain of n   ( n 2 ) is considered in the present paper. We give the necessary and sufficient conditions for the representability of the solution by means of a simple layer hydrodynamic potential, instead of the classical double layer hydrodynamic potential.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 765020, 12 pages.

Dates
First available in Project Euclid: 18 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1366306742

Digital Object Identifier
doi:10.1155/2013/765020

Mathematical Reviews number (MathSciNet)
MR3034912

Zentralblatt MATH identifier
1308.35162

Citation

Cialdea, Alberto; Leonessa, Vita; Malaspina, Angelica. On the Dirichlet Problem for the Stokes System in Multiply Connected Domains. Abstr. Appl. Anal. 2013 (2013), Article ID 765020, 12 pages. doi:10.1155/2013/765020. https://projecteuclid.org/euclid.aaa/1366306742


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References

  • L. Cattabriga, “Su un problema al contorno relativo al sistema di equazioni di Stokes,” Rendiconti del Seminario Matematico della Università di Padova, vol. 31, pp. 308–340, 1961.
  • V. A. Solonnikov, “On estimates of Green's tensors for certain boundary problems,” Doklady Akademii Nauk, vol. 130, pp. 988–991, 1960 (Russian), Translates in Soviet Mathematics Doklady, vol. 1, pp. 128–131, 1960.
  • M. Kohr, “A mixed boundary value problem for the unsteady Stokes system in a bounded domain in $\mathbb{R}_{n}$,” Engineering Analysis with Boundary Elements, vol. 29, no. 10, pp. 936–943, 2005.
  • M. Kohr, “The Dirichlet problems for the Stokes resolvent equations in bounded and exterior domains in ${\mathbb{R}}^{n}$,” Mathematische Nachrichten, vol. 280, no. 5-6, pp. 534–559, 2007.
  • M. Kohr, “The interior Neumann problem for the Stokes resolvent system in a bounded domain in ${\mathbb{R}}^{n}$,” Archives of Mechanics, vol. 59, no. 3, pp. 283–304, 2007.
  • M. Kohr, “Boundary value problems for a compressible Stokes system in bounded domains in ${\mathbb{R}}^{n}$,” Journal of Computational and Applied Mathematics, vol. 201, no. 1, pp. 128–145, 2007.
  • P. Maremonti, R. Russo, and G. Starita, “On the Stokes equations: the boundary value problem,” in Advances in Fluid Dynamics, pp. 69–140, Quaderni di Matematica Aracne, Rome, Italy, 1999.
  • G. Starita and A. Tartaglione, “On the traction problem for the Stokes system,” Mathematical Models & Methods in Applied Sciences, vol. 12, no. 6, pp. 813–834, 2002.
  • A. Cialdea, “On the oblique derivation problem for the Laplace equation, and related topics,” Rendiconti della Accademia Nazionale delle Scienze detta dei XL, vol. 12, no. 1, pp. 181–200, 1988.
  • G. Fichera, “Una introduzione alla teoria delle equazioni integrali singolari,” Rendiconti di Matematica, vol. 17, pp. 82–191, 1958.
  • S. G. Mikhlin and S. Prössdorf, Singular Integral Operators, Springer, Berlin, Germany, 1986.
  • A. Cialdea and G. C. Hsiao, “Regularization for some boundary integral equations of the first kind in mechanics,” Rendiconti della Accademia Nazionale delle Scienze detta dei XL, vol. 19, pp. 25–42, 1995.
  • A. Cialdea, V. Leonessa, and A. Malaspina, “On the Dirichlet and the Neumann problems for Laplace equation in multiply connected domains,” Complex Variables and Elliptic Equations, vol. 57, no. 10, pp. 1035–1054, 2012.
  • A. Malaspina, “Regularization for integral equations of the first kind in the theory of thermoelastic pseudo-oscillations,” Applied Mathematics, Informatics and Mechanics, vol. 9, no. 2, pp. 29–51, 2004.
  • A. Malaspina, “On the traction problem in mechanics,” Archives of Mechanics, vol. 57, no. 6, pp. 479–491, 2005.
  • A. Cialdea, V. Leonessa, and A. Malaspina, “Integral representations for solutions of some BVPs for the Lamé system in multiply connected domains,” Boundary Value Problems, vol. 2011, aticle 53, 2011.
  • A. Malaspina, “Regularization of some integral equations of the first kind,” AIP Conference Proceedings, vol. 1281, pp. 916–919, 2010.
  • A. Malaspina, “Integral representation for the solution of Dirichlet problem for the stokes system,” AIP Conference Proceedings, vol. 1389, pp. 473–476, 2011.
  • A. Cialdea, E. Dolce, A. Malaspina, and V. Nanni, “On an integral equation of the first kind arising in the theory of Cosserat,” submitted.
  • A. Cialdea, “A general theory of hypersurface potentials,” Annali di Matematica Pura ed Applicata, vol. 168, pp. 37–61, 1995.
  • V. D. Kupradze, T. G. Gegelia, M. O. Basheleĭshvili, and T. V. Burchuladze, Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, vol. 25 of North-Holland Series in Applied Mathematics and Mechanics, North-Holland Publishing, Amsterdam, The Netherlands, 1979.
  • O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach Science Publishers, New York, NY, USA, 1969.
  • W. V. D. Hodge, “A dirichlet problem for harmonic functionals, with applications to analytic varities,” Proceedings of the London Mathematical Society, vol. S2-36, no. 1, pp. 257–303, 1934.
  • A. Cialdea, “On the finiteness of the energy integral in elastostatics with non-absolutely continuous data,” Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei IX, vol. 4, no. 1, pp. 35–42, 1993.
  • A. Cialdea, “The multiple layer potential for the biharmonic equation in $n$ variables,” Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei IX, vol. 3, no. 4, pp. 241–259, 1992.