Skip to main content
SpringerLink
Log in

On Friedrichs inequalities for a disk

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

We consider the Friedrichs inequality for functions defined on a disk of unit radius Ω and equal to zero on almost all boundary except for an arc λε of length ɛ, where ɛ is a small parameter. Using the method of matched asymptotic expansions, we construct a two-term asymptotics for the Friedrichs constant \(C(\Omega ,\partial \Omega \backslash \bar \gamma _\varepsilon )\) for such functions and present a strict proof of its validity. We show that \(C(\Omega ,\partial \Omega \backslash \bar \gamma _\varepsilon ) = C(\Omega ,\partial \Omega ) + \varepsilon ^2 C(\Omega ,\partial \Omega )(1 + O(\varepsilon ^{2/7} ))\). The calculation of the asymptotics for the Friedrichs constant is reduced to constructing an asymptotics for the minimum value of the operator −Δ in the disk with Neumann boundary condition on λε and Dirichlet boundary condition on the remaining part of the boundary.

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

Access this article

Log in via an institution

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. O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics (Nauka, Moscow, 1973; Springer-Verlag, Berlin, 1985).

    Google Scholar 

  2. V. P. Mikhailov, Partial Differential Equations (Nauka, Moscow, 1976; Central Books, London, 1979).

    Google Scholar 

  3. R. R. Gadyl’shin, Math. Notes 52(3–4), 1020 (1992).

    Article  MathSciNet  Google Scholar 

  4. R. R. Gadyl’shin, Asymptotic Properties of Solutions to Differential Equations (Bashkirsk. Nauchn. Tsentr Ural’sk. Otdel. Akad. Nauk SSSR, Ufa, 1988), pp. 3–15 [in Russian].

    Google Scholar 

  5. V. S. Vladimirov, Equations of Mathematical Physics (Nauka, Moscow, 1971; Marcel Dekker, New York, 1971.

    Google Scholar 

  6. A. M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems (Nauka, Moscow, 1989; Amer. Math. Soc., Providence, RI, 1991).

    MATH  Google Scholar 

  7. M. Yu. Planida, Math. Notes 75(1–2), 213 (2004).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. M.Akmullah Bashkir State Pedagogical University, ul. Oktyabrskoi Revolyutsii 3a, Ufa, 450000, Russia

    R. R. Gadyl’shin & E. A. Shishkina

Authors
  1. R. R. Gadyl’shin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. A. Shishkina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. R. Gadyl’shin.

Additional information

Dedicated to Arlen Mikhailovich Il’in

Original Russian Text © R.R.Gadyl’shin, E.A. Shishkina, 2012, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2012, Vol. 18, No. 2.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gadyl’shin, R.R., Shishkina, E.A. On Friedrichs inequalities for a disk. Proc. Steklov Inst. Math. 281 (Suppl 1), 44–58 (2013). https://doi.org/10.1134/S0081543813050052

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543813050052

Keywords

Search

Navigation