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Boundary behavior of functions from Sobolev classes defined on domains with exterior peak

Abstract

We establish an invertible characteristic of the boundary behavior of functions from Sobolev spaces defined on a space domain having a vertex of exterior peak on the boundary. The boundary is assumed sufficiently smooth in a neighborhood of the peak vertex. The description of the traces on the boundary is given with the use of weighted Besov spaces.

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References

  1. 1.

    O. V. Besov, V. P. Il’in, and S. M. Nikol’skiĭ, Integral Representations of Functions and Imbedding Theorems (Nauka, Moscow, 1975; Vol. I: V. H. Winston & Sons, Washington, D.C., John Wiley & Sons, New York etc., 1978; Vol. II: V. H. Winston & Sons, Washington, D.C., John Wiley & Sons, New York etc., 1979)

    Google Scholar 

  2. 2.

    D. K. Faddeev, B. Z. Vulikh, and N. N. Ural’tseva, Selected Chapters of Analysis and Higher Algebra (Izdatel’stvo LGU, Leningrad, 1981) [in Russian].

    MATH  Google Scholar 

  3. 3.

    A. Jonsson A. and H. Wallin, “A Whitney extension theorem in L p and Besov spaces,” Ann. Inst. Fourier (Grenoble) 28, 139 (1978).

    Article  MathSciNet  Google Scholar 

  4. 4.

    V. G. Maz’ya, “On functions with finite Dirichlet integral in a domain with a cusp at the boundary,” Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 126, 117 (1983).

    MATH  MathSciNet  Google Scholar 

  5. 5.

    V. G. Maz’ya, Yu. V. Netrusov, and S. V. Poborchiĭ, “Boundary values of functions in Sobolev spaces on certain non-Lipschitzian domains,” Algebra Anal. 11, 141 (1999) [St. Petersburg Math. J. 11, 107 (2000)].

    Google Scholar 

  6. 6.

    V. G. Maz’ya and S. V. Poborchiĭ, “Traces of functions with summable gradient in a domain with a peak summit on the boundary,” Mat. Zametki 45, 57 (1989) [Math. Notes 45, 39 (1989)].

    MATH  Google Scholar 

  7. 7.

    V. G. Maz’ya and S. V. Poborchiĭ, “Boundary traces of functions from Sobolev spaces on a domain with a cusp,” Tr. Inst. Mat. 14, 182 (1989) [Sib. Adv. Math. 1, 75 (1991)].

    Google Scholar 

  8. 8.

    I. M. Pupyshev and M. Yu. Vasil’chik, “An integral representation and boundary behavior of functions defined in a domain with a peak,” Mat. Tr. 13, 23 (2010) [Sib. Adv. Math. 21, 130 (2011)].

    MATH  MathSciNet  Google Scholar 

  9. 9.

    M. Yu. Vasil’chik, “On traces of functions from Sobolev spaces W 1p on domains with non-Lipschitz boundaries,” Trudy Inst. Mat. 14, 9 (1989) [Sib. Adv. Math. 1, 156–199 (1991)].

    MathSciNet  Google Scholar 

  10. 10.

    M. Yu. Vasil’chik, “Boundary properties of functions of the Sobolev space defined in a planar domain with angular points,” Sib. Mat. Zh. 36, 787 (1995) [Sib. Math. J. 36, 677 (1995)].

    Article  MathSciNet  Google Scholar 

  11. 11.

    M. Yu. Vasil’chik, “The boundary behavior of functions of Sobolev spaces defined on a planar domain with a peak vertex on the boundary,” Mat. Tr. 6, 3 (2003) [Sib. Adv. Math. 14, 92 (2004)].

    MATH  MathSciNet  Google Scholar 

  12. 12.

    G. N. Yakovlev, “Boundary Properties of Functions of Class W (l)p on regions with angular points,” Dokl. Akad. Nauk SSSR 140, 73 (1961) [Sov. Math., Dokl. 2, 1177 (1961)].

    MathSciNet  Google Scholar 

  13. 13.

    G. N. Yakovlev, “The Dirichlet problem for domains with non-Lipschitzian boundaries,” Differ. Uravn. 1, 1085 (1965) [Differ. Equations 1, 847 (1965)].

    MATH  Google Scholar 

  14. 14.

    G. N. Yakovlev, “Traces of functions in the space W 1p on piecewise smooth surfaces,” Mat. Sb. 74(116), 526 (1967) [Math. USSR, Sb. 3, 481 (1967)].

    MathSciNet  Google Scholar 

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Correspondence to I. M. Pupyshev.

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Original Russian Text © I. M. Pupyshev and M. Yu. Vasil’chik, 2014, published in Matematicheskie Trudy, 2014, Vol. 17, No. 1, pp. 70–98.

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Pupyshev, I.M., Vasil’chik, M.Y. Boundary behavior of functions from Sobolev classes defined on domains with exterior peak. Sib. Adv. Math. 24, 261–281 (2014). https://doi.org/10.3103/S1055134414040038

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Keywords

  • Sobolev space
  • weighted Besov space
  • peak
  • boundary behavior of functions