Skip to main content
SpringerLink
Log in

Functional Analytic Properties of Multi-Layer Potentials and Boundary Value Problems

  • Chapter
  • First Online:
Multi-Layer Potentials and Boundary Problems

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2063))

  • 1180 Accesses

Abstract

This chapter has a twofold goal. In a first stage, we shall study the Fredholm properties of multi-layer potentials introduced in earlier chapters, while in a second stage we shall proceed to use these results as a tool for establishing the well-posedness of boundary value problems associated with higher-order operators.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. S. Agmon, Multiple layer potentials and the Dirichlet problem for higher order elliptic equations in the plane, I. Comm. Pure Appl. Math. 10, 179–239 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  2. S. Agmon, Remarks on self-adjoint and semi-bounded elliptic boundary value problems, in Proceedings of the International Symposium on Linear Spaces, Jerusalem, 1960 (Jerusalem Academic, Jerusalem, 1960), pp. 1–13

    Google Scholar 

  3. C. Bourgeois, S. Nicaise, Prewavelet approximations for a system of boundary integral equations for plates with free edges on polygons. Math. Meth. Appl. Sci. 21, 1233–1267 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Bourlard, S. Nicaise, Abstract Green formula and applications to boundary integral equations. Numer. Funct. Anal. Optim. 18, 667–689 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Cohen, J. Gosselin, The Dirichlet problem for the biharmonic equation in a C 1 domain in the plane. Indiana Univ. Math. J. 32(5), 635–685 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Cohen, J. Gosselin, Adjoint boundary value problems for the biharmonic equation on C 1 domains in the plane. Arkiv för Math. 23(2), 217–240 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Cwikel, Real and complex interpolation and extrapolation of compact operators. Duke Math. J. 65(2), 333–343 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  8. R. Dautray, J.-L. Lions, in Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 2: Functional and Variational Methods (Springer, Berlin, 2000)

    Google Scholar 

  9. M. Frazier, B. Jawerth, Decomposition of Besov spaces. Indiana Univ. Math. J. 34(4), 777–799 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  10. F. Gazzola, H-.C. Grunau, G. Sweers, in Polyharmonic Boundary Value Problems. Lecture Notes in Mathematics, vol. 1991 (Springer, Berlin, 2010)

    Google Scholar 

  11. J. Giroire, J.-C. Nédélec, A new system of boundary integral equations for plates with free edges. Math. Meth. Appl. Sci. 18(10), 755–772 (1995)

    Article  MATH  Google Scholar 

  12. S. Hofmann, M. Mitrea, M. Taylor, Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains. Int. Math. Res. Not. 2010, 2567–2865 (2010)

    MathSciNet  MATH  Google Scholar 

  13. N. Kalton, M. Mitrea, Stability results on interpolation scales of quasi-Banach spaces and applications. Trans. Am. Math. Soc. 350(10), 3903–3922 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edotion (Springer, Berlin, 1995)

    Google Scholar 

  15. J. Kilty, Z. Shen, A bilinear estimate for biharmonic functions in Lipschitz domains. Math. Ann. 349, 367-394 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. V. Maz’ya, M. Mitrea, T. Shaposhnikova, The Dirichlet problem in Lipschitz domains with boundary data in Besov spaces for higher order elliptic systems with rough coefficients. J. d’Analyse Math. 110(1), 167–239 (2010)

    Google Scholar 

  17. O. Mendez, M. Mitrea, The Banach envelopes of Besov and Triebel-Lizorkin spaces and applications to partial differential equations. J. Fourier Anal. Appl. 6, 503–531 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. D. Mitrea, I. Mitrea, M. Mitrea, A Treatise on the Theory of Elliptic Boundary Value Problems, Singular Integral Operators, and Smoothness Spaces in Rough Domains, book manuscript (2012)

    Google Scholar 

  19. I. Mitrea, M. Mitrea, Higher order elliptic boundary value problems in two-dimensional chord-arc domains, preprint (2012)

    Google Scholar 

  20. I. Mitrea, M. Mitrea, M. Wright, Optimal estimates for the inhomogeneous problem for the bi-Laplacian in three-dimensional Lipschitz domains. J. Math. Sci. 172(1), 24–134 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. M. Mitrea, M. Wright, Boundary value problems for the Stokes system in arbitrary Lipschitz domains, in Astérisque, vol. 344 (Societé Mathématique de France, 2012)

    Google Scholar 

  22. A. Nadai, Theory of Flow and Fracture of Solids, vol. II (McGraw-Hill, New York, 1963)

    Google Scholar 

  23. C. Nazaret, A system of boundary integral equations for polygonal plates with free edges. Math. Meth. Appl. Sci. 21 (2), 165-185 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  24. J. Nečas, in Les Méthodes Directes en Théorie des Équations Elliptiques, Masson et Cie, Éditeurs, Paris (Academia, Éditeurs, Prague, 1967)

    MATH  Google Scholar 

  25. S. Nicaise, Polygonal interface problems for the biharmonic operator. Math. Meth. Appl. Sci. 17, 21–39 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  26. J. Pipher, G.C. Verchota, Maximum principles for the polyharmonic equation on Lipschitz domains. Potential Anal. 4 (6), 615–636 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  27. G. Schmidt, Boundary integral operators for plate bending in domains with corners, Zeitschrift für Analysis und ihre Anwendungen. J. Anal. Appl. 20 (1), 131–154 (2001)

    MATH  Google Scholar 

  28. Z. Shen, The L p Boundary value problems on Lipschitz domains. Adv. Math. 216, 212–254 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. E.M. Stein, in Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, vol. 30 (Princeton University Press, Princeton, 1970)

    Google Scholar 

  30. G. Verchota, The biharmonic Neumann problem in Lipschitz domains. Acta Math. 194, 217–279 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Temple University, Philadelphia, PA, USA

    Irina Mitrea

  2. Department of Mathematics, University of Missouri, Columbia, MO, USA

    Marius Mitrea

Authors
  1. Irina Mitrea
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marius Mitrea
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Mitrea, I., Mitrea, M. (2013). Functional Analytic Properties of Multi-Layer Potentials and Boundary Value Problems. In: Multi-Layer Potentials and Boundary Problems. Lecture Notes in Mathematics, vol 2063. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32666-0_6

Download citation

Publish with us

Policies and ethics

Search

Navigation