Journal of Mathematical Sciences

, Volume 191, Issue 2, pp 150–161 | Cite as

Estimates for solutions to the model Venttsel problem in Campanato spaces

  • A. A. Arkhipova
  • A. A. Lukina

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.


Weak Solution Model Problem Order Derivative Elliptic System Morrey Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. D. Venttsel, “On the boundary conditions for multi-dimensional diffusion processes” [in Russian], Teor. Veroyat. Prim. 4, No. 2, 172–185 (1959).MathSciNetGoogle Scholar
  2. 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. 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. 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).MATHGoogle Scholar
  5. 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).Google Scholar
  6. 6.
    G. M. Troianiello, Elliptic Differential Equations and Obstacle Problems, Plenum Press, New York etc. (1987).MATHGoogle Scholar
  7. 7.
    S. Campanato, “Equasioni ellittiche del secondo ordine e spazi Open image in new window”, Ann. Mat. Pure Appl. 69, No. 4, 321–381 (1965).MathSciNetMATHCrossRefGoogle Scholar
  8. 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).MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

Personalised recommendations