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Sturm–Liouville problems in weighted spaces in domains with non-smooth edges. II


We consider a (generally, noncoercive) mixed boundary value problem in a bounded domain D of Rn for a second order elliptic differential operator A(x, ∂). The differential operator is assumed to be of divergent form in D and the boundary operator B(x, ∂) is of Robin type on ∂D. The boundary of D is assumed to be a Lipschitz surface. Besides, we distinguish a closed subset Y∂D and control the growth of solutions near Y. We prove that the pair (A, B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, the weight function being a power of the distance to the singular set Y. Moreover, we prove the completeness of root functions related to L.

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  1. 1.

    S. Agmon, “On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems”, Comm. Pure Appl.Math. 15, 119 (1962).

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    M. S. Agranovich, “Elliptic operators on closed manifold,” Current Problems of Mathematics, Fundamental Directions, 63, 5 (VINITI, 1990) [in Russian].

    MathSciNet  MATH  Google Scholar 

  3. 3.

    M. S. Agranovich, “On series with respect to root vectors of operators associated with forms having symmetric principal part,” Funct. Anal. Appl. 28 (3), 151 (1994).

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    M. S. Agranovich, “Non-self-adjoint elliptic problems on non-smooth domains,” Russ. J. Math. Phys. 2 (2), 139 (1994).

    MathSciNet  MATH  Google Scholar 

  5. 5.

    M. S. Agranovich, “Spectral problems for second-order strongly elliptic systems in smooth and non-smooth domains,” Russian Math. Surveys 57 (5), 847 (2002).

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    M. S. Agranovich, “Spectral boundary value problems in Lipschitz domains for strongly elliptic systems in Banach spaces Hs p and Bs p,” Funct. Anal. Appl. 42 (4), 249 (2008).

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    M. S. Agranovich, “Spectral problems in Lipschitz domains,” Modern Mathematics, Fundamental Trends 39, 11 (2011).

    MathSciNet  MATH  Google Scholar 

  8. 8.

    M. S. Agranovich, “Strongly elliptic second order systems with boundary conditions on a non-closed Lipschitz surface,” Funct. Anal. Appl. 45 (1), 1 (2011).

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    M. S. Agranovich, “Mixed problems in a Lipschitz domain for strongly elliptic second order systems,” Funct. Anal. Appl. 45 (2), 81 (2011).

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    L. A. Aizenberg and A. M. Kytmanov, “On the possibility of holomorphic continuation to a domain of functions given on a part of its boundary,” Math. Sbornik 182 (4), 490 (1991).

    Google Scholar 

  11. 11.

    F. E. Browder, “On the eigenfunctions and eigenvalues of the general elliptic differential operator,” Proc. Nat. Acad. Sci. USA 39, 433 (1953).

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    F. E. Browder, “Estimates and existence theorems for elliptic boundary value problems,” Proc. Nat. Acad. Sci. USA 45, 365 (1959).

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    F. E. Browder, “On the spectral theory of strongly elliptic differential operators,” Proc. Nat. Acad. Sci. USA 45, 1423 (1959).

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    V. I. Burenkov, Sobolev Spaces on Domains, Teubner-Texte zurMathematik 137 (B.G. Teubner, Stuttgart, 1998).

    Book  MATH  Google Scholar 

  15. 15.

    N. Dunford and J. T. Schwartz, Linear Operators, Vol. II, Self-Adjoint Operators in Hilbert Space (Intersci. Publ., New York, 1963).

    MATH  Google Scholar 

  16. 16.

    Yu. Egorov, V. Kondratiev, and B. W. Schulze, “Completeness of eigenfunctions of an elliptic operator on a manifold with conical points,” Russ. J. Math. Phys. 8, (3), 267 (2001).

    MathSciNet  MATH  Google Scholar 

  17. 17.

    I. Ts. Gokhberg and M. G. Krein, Introduction to the Theory of Linear Nonself-adjoint Operators in Hilbert Spaces (AMS, Providence, R.I., 1969).

    Google Scholar 

  18. 18.

    I. Ts. Gokhberg and E. I. Sigal, “An operator generalization of the logarithmic residue theorem and the theorem of Rouché,” Math. Sbornik 13, 603 (1971).

    Article  Google Scholar 

  19. 19.

    P. Hartman, Ordinary Differential Equation (John Wiley and Sons, New York, 1964).

    MATH  Google Scholar 

  20. 20.

    L. I. Hedberg and T. H. Wolff, “Thin sets in nonlinear potential theory,” Ann. Inst. Fourier (Grenoble) 33 (4), 161 (1983).

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    L. Hörmander, “Hypoelliptic second order differential equations,” Acta Math. 119, 147 (1967).

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    M.V. Keldysh, “On the characteristic values and characteristic functions of certain classes of non-self-adjoint equations,” Dokl. AN SSSR 77, 11 (1951) [in Russian].

    MathSciNet  Google Scholar 

  23. 23.

    M. V. Keldysh, “On the completeness of eigenfunctions of some classes of non-self-adjoint linear operators,” Russian Math. Surveys 26 (4), 15 (1971) [in Russian].

    Article  MATH  Google Scholar 

  24. 24.

    J. J. Kohn, “Subellipticity of the ∂-Neumann problem on pseudoconvex domains: sufficient conditions,” Acta Math. 142 (1,2), 79 (1979).

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    V. A. Kondrat’ev, “Completeness of the systems of root functions of elliptic operators in Banach spaces,” Russ. J. Math. Phys. 6 (10), 194 (1999).

    MathSciNet  MATH  Google Scholar 

  26. 26.

    O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Equations of Elliptic Type (Nauka, Moscow, 1973) [in Russian].

    Google Scholar 

  27. 27.

    J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems und Applications. Vol. 1 (Springer-Verlag, Berlin, 1972).

    Book  MATH  Google Scholar 

  28. 28.

    V. P. Mikhailov, Partial Differential Equations (Nauka, Moscow, 1976) [in Russian].

    MATH  Google Scholar 

  29. 29.

    B.V. Paltsev, “Mixed problemswith non-homogeneous boundary conditions in Lipschitz domains for secondorder elliptic equations with a parameter,” Math. Sbornik, 187 (4), 59 (1996).

    MathSciNet  Google Scholar 

  30. 30.

    B. A. Plamenevskiĭ, Algebras of Pseudodifferential Operators (Nauka, Moscow, 1986) [in Russian].

    MATH  Google Scholar 

  31. 31.

    A. Polkovnikov and A. Shlapunov, “On the spectral properties of a noncoercive mixed problem associated with ∂-operator,” J. Siberian Fed. Uni. 6 (2), 247 (2013) [in Russian].

    Google Scholar 

  32. 32.

    A. A. Shlapunov, “Spectral decomposition of Green’s integrals and existence of W s,2-solutions of matrix factorizations of the Laplace operator in a ball,” Rend. Sem.Mat. Univ. Padova 96, 237 (1996).

    MathSciNet  MATH  Google Scholar 

  33. 33.

    A. A. Shlapunov and N. N. Tarkhanov, “Duality by reproducing kernels,” Int. J. of Math. and Math. Sc. 6, 327 (2003).

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    A. A. Shlapunov and N. N. Tarkhanov, “On completeness of root functions of Sturm-Liouville problems with discontinuous boundary operators,” J. of Differential Equations 255, 3305 (2013).

    MathSciNet  Article  MATH  Google Scholar 

  35. 35.

    A. A. Shlapunov and N. N. Tarkhanov, “Sturm-Liouville Problems in Weighted Spaces in Domains with Non-Smooth Edges. II,” Siberian Adv. Math. 26 (1), 30 (2016).

    MathSciNet  Article  Google Scholar 

  36. 36.

    L.N. Slobodetskiĭ, “generalized spaces of S.L. Sobolev and their applications to boundary problems for partial differential equations,” Science Notes of Leningr. Pedag. Institute 197, 54 (1958) [in Russian].

    Google Scholar 

  37. 37.

    N. Tarkhanov, “On the root functions of general elliptic boundary value problems,” Compl. Anal.Oper. Theory 1, 115 (2006).

    MathSciNet  Article  MATH  Google Scholar 

  38. 38.

    A. N. Tikhonov and A. A. Samarskiĭ, Equations of Mathematical Physics (Nauka, Moscow, 1972) [in Russian].

    MATH  Google Scholar 

  39. 39.

    B. L. Van der Waerden, Algebra (Springer-Verlag, Berlin, 1967).

    MATH  Google Scholar 

  40. 40.

    S. Zaremba, “Sur un problème mixte relatif à l’équation de Laplace,” Bull. Acad. Sci. Cracovie, 314 (1910).

    Google Scholar 

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Correspondence to A. A. Shlapunov.

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Original Russian Text ©A. A. Shlapunov and N. Tarkhanov, 2015, published in Matematicheskie Trudy, 2015, Vol. 18, No. 2, pp. 133–204.

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Shlapunov, A.A., Tarkhanov, N.N. Sturm–Liouville problems in weighted spaces in domains with non-smooth edges. II. Sib. Adv. Math. 26, 247–293 (2016).

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  • mixed problems
  • noncoercive boundary conditions
  • elliptic operators
  • root functions
  • weighted Sobolev spaces