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LP-Regularity for Poincaré Problem and Applications

  • Dian K. Palagachev
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 79)

Abstract

We improve the results from [10] on strong solvability and uniqueness for the oblique derivative problem
$$ \left\{ \begin{gathered} a^{ij} \left( x \right)D_{ij} u + b^i \left( x \right)D_i u + c\left( x \right)u = f\left( x \right) a.a. in \Omega , \hfill \\ \partial u/\partial \ell + \sigma \left( x \right)u = \varphi \left( x \right) on \partial \Omega , \hfill \\ \end{gathered} \right. $$
extending them to Sobolev’s space W 2,p (Ω), for any p > 1. The vector field ℓ(x) tangent to Ω at the points of ɛΩ and directed outwards Ω on Ω\ɛ.

Key words

Uniformly elliptic operator Poincaré problem Strong solutions 

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References

  1. [1]
    F. Chiarenza, M. Frasca and P. Longo, W 2,p-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc. 336 (1993), 841–853.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Y.V. Egorov, Linear Differential Equations of Principal Type, Contemporary Soviet Mathematics, New York, 1986.zbMATHGoogle Scholar
  3. [3]
    Y.V. Egorov and V. Kondrat’ev, The oblique derivative problem, Math. USSR Sbornik 7 (1969). 139–169.zbMATHCrossRefGoogle Scholar
  4. [4]
    D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, Berlin, 1983.zbMATHGoogle Scholar
  5. [5]
    P. Guan, Hölder regularity of subelliptic pseudodifferential operators, Duke Math. J. 60 (1990), 563–598.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    P. Guan and E. Sawyer, Regularity estimates for the oblique derivative problem, Ann. Math. 137 (1993), 1–70.CrossRefMathSciNetGoogle Scholar
  7. [7]
    P. Guan and E. Sawyer, Regularity estimates for the oblique derivative problem on non-smooth domains I, Chinese Ann. Math., Ser. B 16 (1995), No. 3, 1–26; II ibid. 17 (1996), No. 1, 1–34.MathSciNetGoogle Scholar
  8. [8]
    L. Hörmander, Pseudodifferential operators and non-elliptic boundary value problems, Ann. Math. 83 (1966), 129–209.CrossRefGoogle Scholar
  9. [9]
    A. Maugeri, D.K. Palagachev and L.G. Softova, Elliptic and Parabolic Equations with Discontinuous Coefficients, Wiley-VCH, Berlin, 2000.zbMATHGoogle Scholar
  10. [10]
    A. Maugeri, D.K. Palagachev and C. Vitanza, A singular boundary value problem for umformly elliptic operators, J. Math. Anal. Appl. 263 (2001), 33–48.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    V. Maz’ya and B.P. Paneah, Degenerate elliptic pseudodifferential operators and oblique derivative problem, Trans. Moscow Math. Soc. 31 (1974), 247–305.zbMATHGoogle Scholar
  12. [12]
    A. Melin and J. Sjöstrand, Fourier Integral Operators with Complex-Valued Phase Functions, in: Lect. Notes Math., Vol. 459, pp. 120–223, Springer-Verlag, Berlin, 1975.Google Scholar
  13. [13]
    B.P. Paneah, The Oblique Derivative Problem. The Poincaré Problem, Wiley-VCH, Berlin, 2000.zbMATHGoogle Scholar
  14. [14]
    H. Poincaré, Lecons de Méchanique Céleste, Tome III, Théorie de Marées, Gauthiers-Villars, Paris, 1910.Google Scholar
  15. [15]
    P.R. Popivanov and D.K. Palagachev, The Degenerate Oblique Derivative Problem for Elliptic and Parabolic Equations, Wiley-VCH (Akademie-Verlag), Berlin, 1997.zbMATHGoogle Scholar
  16. [16]
    N.S. Trudinger, Nonlinear Second Order Elliptic Equations, in: Lecture Notes of Math. Inst. of Nankai Univ., Tianjin, China, 1986.Google Scholar
  17. [17]
    B. Winzell, A boundary value problem with an oblique derivative, Commun. Partial Differ. Equations 6 (1981), 305–328.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Dian K. Palagachev
    • 1
  1. 1.Dept. of MathematicsTechnical University of BariBariItaly

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