Skip to main content
Log in

W 2,p-a priori estimates for the emergent Poincaré Problem

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

We derive W 2,p(Ω)-a priori estimates with arbitrary p ∈(1, ∞), for the solutions of a degenerate oblique derivative problem for linear uniformly elliptic operators with low regular coefficients. The boundary operator is given in terms of directional derivative with respect to a vector field ℓ that is tangent to ∂Ω at the points of a non-empty set ε ⊂ ∂Ω and is of emergent type on ∂Ω.

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. Egorov Y.V. and Kondrat’ev V. (1969). The oblique derivative problem. Math. USSR Sbornik 7: 139–169

    Article  Google Scholar 

  2. Gilbarg D. and Trudinger N.S. (1983). Elliptic Partial Differential Equations of Second Order. Springer, Berlin, Heidelberg, New York

    Google Scholar 

  3. Guan P. and Sawyer E. (1993). Regularity estimates for the oblique derivative problem. Ann. Math. 137: 1–70

    Article  Google Scholar 

  4. Guan, P., Sawyer, E.: Regularity estimates for the oblique derivative problem on non-smooth domains I. Chin. Ann. Math., Ser. B 16, 1–26 (1995); II, ibid. 17, 1–34 (1996)

  5. Hörmander L. (1966). Pseudodifferential operators and non-elliptic boundary value problems. Ann. Math. 83: 129–209

    Article  Google Scholar 

  6. Kufner A., John O. and Fučík S. (1977). Function Spaces. Noordhoff, Leyden

    Google Scholar 

  7. Maugeri A., Palagachev D.K. and Softova L.G. (2000). Elliptic and Parabolic Equations with Discontinuous Coefficients. Math. Res., vol. 109.. Wiley–VCH, Berlin

    Google Scholar 

  8. Maz’ya V. (1972). On a degenerating problem with directional derivative. Math. USSR Sbornik 16: 429–469

    Article  Google Scholar 

  9. Maz’ya V. and Paneah B.P. (1974). Degenerate elliptic pseudodifferential operators and oblique derivative problem. Trans. Moscow Math. Soc. 31: 247–305

    Google Scholar 

  10. Melin A. and Sjöstrand J. (1976). Fourier integral operators with complex phase functions and parametrix for an interior boundary value problem. Commun. Partial Differ. Equations 1: 313–400

    Article  Google Scholar 

  11. Palagachev D.K. (1992). The tangential oblique derivative problem for second order quasilinear parabolic operators. Commun. Partial Differ. Equations 17: 867–903

    Article  Google Scholar 

  12. Palagachev D.K. (2005). The Poincaré problem in L p-Sobolev spaces I: codimension one degeneracy. J. Funct. Anal. 229: 121–142

    Article  Google Scholar 

  13. Palagachev, D.K.: Neutral Poincaré Problem in L p-Sobolev Spaces: regularity and fredholmness. Int. Math. Res. Notices 2006, Article ID 87540, p. 31 (2006)

  14. Palagachev, D.K.: The Poincaré problem in L p-Sobolev spaces II: full dimension degeneracy. Commun. Partial Differ. Equations (2007) (to appear)

  15. Paneah B.P. (1978). On a problem with oblique derivative. Soviet Math. Dokl. 19: 1568–1572

    Google Scholar 

  16. Paneah B.P. (2000). The Oblique Derivative Problem. The Poincaré Problem, Math. Topics, vol. 17, Wiley–VCH, Berlin

    Google Scholar 

  17. Poincaré H. (1910). Lecons de Méchanique Céleste, Tome III, Théorie de Marées. Gauthiers–Villars, Paris

    Google Scholar 

  18. Popivanov P.R. and Kutev N.D. (1989). The tangential oblique derivative problem for nonlinear elliptic equations. Commun. Partial Differ. Equations 14: 413–428

    Article  Google Scholar 

  19. Popivanov P.R. and Kutev N.D. (2005). Viscosity solutions to the degenerate oblique derivative problem for fully nonlinear elliptic equations. Math. Nachr. 278: 888–903

    Article  Google Scholar 

  20. Popivanov P.R. and Palagachev D.K. (1997). The Degenerate Oblique Derivative Problem for Elliptic and Parabolic Equations. Math. Res., vol. 93. Akademie–Verlag, Berlin

    Google Scholar 

  21. Winzell B. (1977). The oblique derivative problem I. Math. Ann. 229: 267–278

    Article  Google Scholar 

  22. Winzell B. (1981). A boundary value problem with an oblique derivative. Commun. Partial Differ. Equations 6: 305–328

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Dian K. Palagachev.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Palagachev, D.K. W 2,p-a priori estimates for the emergent Poincaré Problem. J Glob Optim 40, 305–318 (2008). https://doi.org/10.1007/s10898-007-9175-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-007-9175-8

Keywords

Mathematics Subject Classification (2000)

Navigation