Skip to main content
Log in

Estimates for solutions to the model Venttsel problem in Campanato spaces

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

We consider a linear elliptic system of second order equations with constant coefficients in a neighborhood of the plane part of the boundary. The boundary condition contains the conormal derivative and the second order operator containing the tangent derivatives (a Venttsel type problem). For a weak solution we obtain the fundamental Campanato estimates. We clarify the dependence of the smoothness of a weak solution to the corresponding inhomogeneous problem in Campanato spaces on the smoothness of the right-hand sides of the system and boundary conditions. Bibliography: 8 titles.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. D. Venttsel, “On the boundary conditions for multi-dimensional diffusion processes” [in Russian], Teor. Veroyat. Prim. 4, No. 2, 172–185 (1959).

    MathSciNet  Google Scholar 

  2. D. A. Apushkinskaya and A. I. Nazrov, “On the quasilinear stationary Venttsel boundary value problem” [in Russian], Zap. Nauchn. Semin. POMI 221, 20–29 (1995); English transl.: J. Math. Sci., New York 87, No. 2, 3277–3283 (1997).

    Google Scholar 

  3. V. V. Luk’yanov and A. I. Nazarov, “Solving the Venttsel’ problem for the Laplace and Helmholtz equations with the help of iterated potentials” [in Russian], Zap. Nauchn. Semin. POMI 250, 203–218, (1998); English transl.: J. Math. Sci., New York 102, No. 4, 4265–4274 (2000).

    Google Scholar 

  4. A. I. Nazarov, “On the nonstationary two-phase Venttsel problem in the transversal case” [in Russian], Probl. Mat. Anal. 28, 71–82 (2004); English transl.: J. Math. Sci., New York 122, No. 3, 3251–3264 (2004).

    MATH  Google Scholar 

  5. A. A. Arkhipova, Regularity of Weak Solutions to Boundary Value Problems for Linear Equations and Systems of Elliptic Type [in Russian], St. Petersb. State Univ. Press, St. Petersburg (198).

  6. G. M. Troianiello, Elliptic Differential Equations and Obstacle Problems, Plenum Press, New York etc. (1987).

    MATH  Google Scholar 

  7. S. Campanato, “Equasioni ellittiche del secondo ordine e spazi ”, Ann. Mat. Pure Appl. 69, No. 4, 321–381 (1965).

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Giaquinta and 1 G. Modica, “Non-linear systems of the type of the stationary Navie-Stokes systems”, J. Reine Angew. Math. 330, 173–214, (1982).

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. A. Arkhipova.

Additional information

Translated from Problemy Matematicheskogo Analiza 70, May 2013, pp. 47–56.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Arkhipova, A.A., Lukina, A.A. Estimates for solutions to the model Venttsel problem in Campanato spaces. J Math Sci 191, 150–161 (2013). https://doi.org/10.1007/s10958-013-1309-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-013-1309-6

Keywords

Navigation