Journal of Integral Equations and Applications

Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, I: Equivalence and invertibility

O. Chkadua, S.E. Mikhailov, and D. Natroshvili

Full-text: Open access

Article information

Source
J. Integral Equations Applications Volume 21, Number 4 (2009), 499-543.

Dates
First available in Project Euclid: 31 December 2009

Permanent link to this document
https://projecteuclid.org/euclid.jiea/1262271458

Digital Object Identifier
doi:10.1216/JIE-2009-21-4-499

Mathematical Reviews number (MathSciNet)
MR2577510

Zentralblatt MATH identifier
1204.65139

Keywords
Partial differential equation variable coefficients mixed problem parametrix boundary-domain integral equations pseudo-differential equations existence uniqueness invertibility

Citation

Chkadua, O.; Mikhailov, S.E.; Natroshvili, D. Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, I: Equivalence and invertibility. J. Integral Equations Applications 21 (2009), no. 4, 499--543. doi:10.1216/JIE-2009-21-4-499. https://projecteuclid.org/euclid.jiea/1262271458.


Export citation

References

  • C.A. Brebbia, J.C.F. Telles and L.C. Wrobel, Boundary element techniques, Springer-Verlag, Berlin, 1984.
  • T.V. Burchuladze and T.G. Gegelia, Development of the potential methods in the theory of elasticity, Metsniereba, Tbilisi, 1985 (in Russian).
  • L. Castro, R. Duduchava and F.-O. Speck, Localization and minimal normalization of some basic mixed boundary value problems, in Factorization, singular operators and related problems, Proc. Conf. in Honour of Professor Georgii Litvinchuk at Funchal, Portugal 2002, S. Samko, et al., eds., Kluwer, Dordrecht, 2003, 73-100.
  • O. Chkadua, S. Mikhailov and D. Natroshvili, Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, II, J. Integral Equations Appl., to appear.
  • --------, About analysis of some localized boundary-domain integral equations for a variable-coefficient BVP, in Advances in boundary integral methods, Proc. 6th UK Conference on Boundary Integral Methods, J. Trevelyan, ed., Durham University Publ., UK, %ISBN 978-0 -9535558-3-3, 2007, 291-302.
  • C. Constanda, Direct and indirect boundary integral equation methods, Chapman & Hall/CRC, Boca Raton, 1999.
  • M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM J. Math. Anal. 19 (1988), 613-626.
  • M. Costabel and E.P. Stephan, An improved boundary element Galerkin method for three-dimensional crack problems, Integral Equations Operator Theory 10 (1987), 467-504.
  • M. Costabel and W.L. Wendland, Strong ellipticity of boundary integral operators, J. reine angew. Math. 372 (1986), 34-63.
  • 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), 795-818.
  • R. Dautray and J.L. Lions, Mathematical analysis and numerical methods for science and technology, Vol. 4, Integral Equations and Numerical Methods, Springer, Berlin, 1990.
  • R. Duduchava, D. Natroshvili and E. Shargorodsky, Boundary value problems of the mathematical theory of cracks, Proc. I. Vekua Institute of Applied Mathematics, Tbilisi State University 39 (1990), 68-84.
  • --------, Basic boundary value problems of thermoelasticity for anisotropic bodies with cuts, I & II, Georgian Math. Journal 2 (1995), 123-140, 259-276.
  • G. Eskin, Boundary value problems for elliptic pseudodifferential equations, Transl. Math. Monographs 52, American Mathematical Society, Providence, RI, 1981.
  • P. Grisvard, Boundary value problems in nonsmooth domains, Pittman, London, 1985.
  • D. Hilbert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen, Teubner, Leipzig, 1912.
  • G.C. Hsiao and W.L. Wendland, A finite element method for some integral equations of the first kind, J. Math. Anal. Appl. 58 (1977), 449-481.
  • V.D. Kupradze, T.G. Gegelia, M.O. Basheleishvili and T.V. Burchuladze, Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity, North-Holland Publishing Company, Amsterdam, 1979.
  • E.E. Levi, I problemi dei valori al contorno per le equazioni lineari totalmente ellittiche alle derivate parziali, Mem. Soc. Ital. dei Sci. %40 (1909), 16 (1909), 1-112.
  • J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Vol. 1, Springer, Berlin, 1972.
  • V.G. Maz'ya, Boundary integral equations, in Encyclopedia of mathematical sciences, Vol. 27, R.V. Gamkrelidze, ed., Springer-Verlag, Heidelberg, 1991.
  • --------, On the theory of potential for the Lamé system in a domain with piece-wise smooth boundary, in Partial differential equations and their applications, Proc. All-Union Symposium in Tbilisi, %21-23 April 1982. Tbilisi Univ. Publ. 1986, 123-129 (in Russian).
  • W. McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000.
  • S.E. Mikhailov, On an integral equation of some boundary value problems for harmonic functions in plane multiply connected domains with nonregular boundary, Matematicheskii Sbornik 121 (1983), 533-544; Math. of the USSR-Sbornik 49 (1984), 525-536 (in English).
  • --------, Finite-dimensional perturbations of linear operators and some applications to boundary integral equations, Engineering Anal. Boundary Elements 23 (1999), 805-813.
  • --------, Localized boundary-domain integral formulation for problems with variable coefficients, Engineer. Anal. Boundary Elements 26 (2002), 681-690.
  • --------, Analysis of extended boundary-domain integral and integro-differential equations of some variable-coefficient BVP, in Advances in boundary integral methods, Proc. 5th UK Conference on Boundary Integral Methods, Ke Chen, ed., University of Liverpool Publ., UK, %ISBN 0 906370 39 6, 2005, 106-125.
  • --------, Analysis of united boundary-domain integro-differential and integral equations for a mixed BVP with variable coefficient, Math. Methods Appl. Sci. 29 (2006), 715-739.
  • S.G. Mikhlin and S. Prössdorf, Singular integral operators, Springer, Berlin, 1986.
  • C. Miranda, Partial differential equations of elliptic type, 2nd ed., Springer, Berlin, 1970.
  • D. Mitrea, M. Mitrea and J. Pipher, Vector potential theory on non-smooth domains in $\r^3$ and applications to electromagnetic scattering, J. Fourier Anal. Appl. 3 (1997), 1419-1448.
  • D. Mitrea, M. Mitrea and M. Taylor, Layer potentials, the Hodge Laplacian, and global boundary value problems in non-smooth Riemannian manifolds, Memoirs Amer. Math. Soc. 150, (2001), vii+120 pages.
  • D. Natroshvili, O. Chkadua and E. Shargorodsky, Mixed boundary value problems of the anisotropic elasticity, Proc. I. Vekua Inst. Appl. Math. Tbilisi State Univ. 39 (1990), 133-181 (in Russian).
  • J. Nečas, Méthodes Directes en Théorie des Équations Élliptique, Masson, Paris, 1967.
  • A. Pomp, The boundary-domain integral method for elliptic systems. With applications in shells, Lecture Notes Math. 1683, Springer, Berlin, 1998.
  • R.T. Seeley, Singular integrals and boundary value problems, Amer. J. Math., 88 (1966), 781-809.
  • O. Steinbach and W.L. Wendland, On C. Neumann's method for second-order elliptic systems in domains with non-smooth boundaries, J. Math. Anal. Appl. 262 (2001) 733-748.
  • E.P. Stephan, Boundary integral equations for screen problems in $\r^3$, Integral Equations Operator Theory 10 (1987), 236-257.
  • --------, Boundary integral equations for mixed boundary value problems in $\r^3$, Math. Nachr. 134 (1987), 21-53.
  • H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland, Amsterdam, 1978.
  • G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains, J. Functional Anal. 59 (1984), 572-611.
  • W.L. Wendland, E. Stephan and G.C. Hsiao, On the integral equation method for the plane mixed boundary value problem of the Laplacian, Math. Meth. Appl. Sci. 1 (1979), 265-321.