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.
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Egorov Y.V. and Kondrat’ev V. (1969). The oblique derivative problem. Math. USSR Sbornik 7: 139–169
Gilbarg D. and Trudinger N.S. (1983). Elliptic Partial Differential Equations of Second Order. Springer, Berlin, Heidelberg, New York
Guan P. and Sawyer E. (1993). Regularity estimates for the oblique derivative problem. Ann. Math. 137: 1–70
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)
Hörmander L. (1966). Pseudodifferential operators and non-elliptic boundary value problems. Ann. Math. 83: 129–209
Kufner A., John O. and Fučík S. (1977). Function Spaces. Noordhoff, Leyden
Maugeri A., Palagachev D.K. and Softova L.G. (2000). Elliptic and Parabolic Equations with Discontinuous Coefficients. Math. Res., vol. 109.. Wiley–VCH, Berlin
Maz’ya V. (1972). On a degenerating problem with directional derivative. Math. USSR Sbornik 16: 429–469
Maz’ya V. and Paneah B.P. (1974). Degenerate elliptic pseudodifferential operators and oblique derivative problem. Trans. Moscow Math. Soc. 31: 247–305
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
Palagachev D.K. (1992). The tangential oblique derivative problem for second order quasilinear parabolic operators. Commun. Partial Differ. Equations 17: 867–903
Palagachev D.K. (2005). The Poincaré problem in L p-Sobolev spaces I: codimension one degeneracy. J. Funct. Anal. 229: 121–142
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)
Palagachev, D.K.: The Poincaré problem in L p-Sobolev spaces II: full dimension degeneracy. Commun. Partial Differ. Equations (2007) (to appear)
Paneah B.P. (1978). On a problem with oblique derivative. Soviet Math. Dokl. 19: 1568–1572
Paneah B.P. (2000). The Oblique Derivative Problem. The Poincaré Problem, Math. Topics, vol. 17, Wiley–VCH, Berlin
Poincaré H. (1910). Lecons de Méchanique Céleste, Tome III, Théorie de Marées. Gauthiers–Villars, Paris
Popivanov P.R. and Kutev N.D. (1989). The tangential oblique derivative problem for nonlinear elliptic equations. Commun. Partial Differ. Equations 14: 413–428
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
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
Winzell B. (1977). The oblique derivative problem I. Math. Ann. 229: 267–278
Winzell B. (1981). A boundary value problem with an oblique derivative. Commun. Partial Differ. Equations 6: 305–328
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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
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DOI: https://doi.org/10.1007/s10898-007-9175-8
Keywords
- Uniformly elliptic operator
- Poincaré problem
- Emergent vector field
- Strong solution
- A priori estimates
- L p-Sobolev spaces