Functional Analysis and Its Applications

, Volume 42, Issue 4, pp 249–267

Spectral boundary value problems in Lipschitz domains for strongly elliptic systems in Banach spaces Hpσ and Bpσ

Article

Abstract

In a bounded Lipschitz domain, we consider a strongly elliptic second-order equation with spectral parameter without assuming that the principal part is Hermitian. For the Dirichlet and Neumann problems in a weak setting, we prove the optimal resolvent estimates in the spaces of Bessel potentials and the Besov spaces. We do not use surface potentials. In these spaces, we derive a representation of the resolvent as a ratio of entire analytic functions with sharp estimates of their growth and prove theorems on the completeness of the root functions and on the summability of Fourier series with respect to them by the Abel-Lidskii method. Preliminarily, such questions for abstract operators in Banach spaces are discussed. For the Steklov problem with spectral parameter in the boundary condition, we obtain similar results. We indicate applications of the resolvent estimates to parabolic problems in a Lipschitz cylinder. We also indicate generalizations to systems of equations.

Key words

strong ellipticity Lipschitz domain potential space Besov space weak solution optimal resolvent estimate determinant of a compact operator completeness of root functions Abel-Lidskii summability parabolic semigroup 

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References

  1. [1]
    Sh. Agmon, “On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems,” Comm. Pure Appl. Math., 15:2 (1962), 119–147.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    M. S. Agranovich, “Spectral properties of diffraction problems,” A supplement to the book: N. N. Voitovich, B. Z. Katsenelenbaum, and A. N. Sivov, Generalized Method of Eigenoscillations in Diffraction Theory, Nauka, Moscow, 1977, 289–416; English revised edition: M. S. Agranovich, B. Z. Katsenelenbaum, A. N. Sivov, and N. N. Voitovich, Generalized Method of Eigenoscillations in Diffraction Theory, Wiley-VCH, Berlin etc., 1999, Chapter V.Google Scholar
  3. [3]
    M. S. Agranovich, “Spectral problems for second-order strongly elliptic systems in smooth and non-smooth domains,” Uspekhi Mat. Nauk, 57:5 (2002), 3–78; English transl.: Russian Math. Surveys, 2002, No. 5, 847–920.MathSciNetGoogle Scholar
  4. [4]
    M. S. Agranovich, “On a mixed Poincaré-Steklov type spectral problem in a Lipschitz domain,” Russian J. Math. Phys., 13:3 (2006), 239–244.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    M. S. Agranovich, “Regularity of variational solutions to linear boundary value problems in Lipschitz domains,” Funkts. Anal. Prilozhen., 40:4 (2006), 83–103; English transl.: Functional Anal. Appl., 40:4 (2006), 313–329.MathSciNetGoogle Scholar
  6. [6]
    M. S. Agranovich, “To the theory of the Dirichlet and Neumann problems for linear strongly elliptic systems in Lipschitz domains,” Funkts. Anal. Prilozhen., 41:4 (2007), 1–21; English transl.: Functional Anal. Appl., 40:4 (2007), 247–263.MathSciNetGoogle Scholar
  7. [7]
    M. S. Agranovich, “Remarks on potential spaces and Besov spaces in a Lipschitz domain and on Whitney arrays on its boundary,” Russian J. Math. Phys., 15:2 (2008), 146–155.CrossRefMathSciNetGoogle Scholar
  8. [8]
    M. S. Agranovich and M. I. Vishik, “Elliptic problems with parameter and parabolic problems of general form,” Uspekhi Mat. Nauk, 10:3 (1964), 53–161; English transl.: Russian Math. Surveys, 19:3 (1964), 53–157.Google Scholar
  9. [9]
    J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-New York, 1976.MATHGoogle Scholar
  10. [10]
    M. S. Birman and M. Z. Solomyak, “Piecewise-polynomial approximations of functions of the classes W pα,” Mat. Sb., 73:3 (1967), 331–355.MathSciNetGoogle Scholar
  11. [11]
    M. S. Birman and M. Z. Solomyak, “Spectral asymptotics of nonsmooth elliptic operators, I, II,” Trudy Moskov. Mat. Obshch., 27 (1972), 3–52; 28 (1973), 3–34; English transl.: Trans. Moscow Math. Soc., 27 (1975), 3–52; 28 (1975), 1–32.MATHGoogle Scholar
  12. [12]
    M. S. Birman and M. Z. Solomyak, “Quantitative analysis in Sobolev embeddings theorems and applications to spectral theory,” in: 10th Math. School, Kiev, 1974, 5–189; English transl.: Amer. Math. Soc. Transl. (2), vol. 114, Amer. Math. Soc., Providence, RI, 1980.Google Scholar
  13. [13]
    R. M. Brown and Z. Shen, “A note on boundary value problems for the heat equation in Lipschitz cylinders,” Proc. Amer. Math. Soc., 119:2 (1993), 585–594.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    J. Burgoyne, “Denseness of the generalized eigenvectors of a discrete operator in a Banach space,” J. Operator Theory, 33 (1995), 279–297.MATHMathSciNetGoogle Scholar
  15. [15]
    N. Dunford and J. T. Schwartz, Linear Operators, Part II, Interscience Publishers, New York-London, 1963.MATHGoogle Scholar
  16. [16]
    D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford Univ. Press, Oxford, 1987.MATHGoogle Scholar
  17. [17]
    D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, and Differential Operators, Cambridge Univ. Press, Cambridge, 1996.Google Scholar
  18. [18]
    I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Non-Selfadjoint Operators in a Hilbert Space, Nauka, Moscow, 1965; English transl.: Transl. Math. Monographs, vol. 18, Amer. Math. Soc., Providence, RI, 1968.Google Scholar
  19. [19]
    A. Grothendieck, “Produits tenzoriels topologiques et espaces nucléares,” Mem. Amer. Math. Soc., 16 (1955).Google Scholar
  20. [20]
    A. Grothendieck, “La théorie de Fredholm,” Bull. Soc. Math. France, 84 (1956), 319–384.MATHMathSciNetGoogle Scholar
  21. [21]
    S. Janson, P. Nilsson, and J. Peetre, “Notes on Wolff’s note on interpolation spaces,” Proc. London Math. Soc., 48:2 (1984), 283–299.MATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    A. Jonsson and H. Wallin, Function Spaces on Subsets of ℝn, Mathematical Reports, vol. 2, part 1, Harwood Academic Publishers, London etc., 1984.Google Scholar
  23. [23]
    H. König, Eigenvalue Distribution of Compact Operators, Operator Theory: Advances and Applications, vol. 16, Birkhäuser, Basel etc., 1986.Google Scholar
  24. [24]
    V. A. Kozlov, V. G. Maz’ya, and J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities, Amer. Math. Soc., Providence, RI, 1997.MATHGoogle Scholar
  25. [25]
    P. D. Lax and A. N. Milgram, “Parabolic equations,” in: Contributions to the Theory of Partial Differential Equations, Ann. of Math. Studies, vol. 33, Princeton Univ. Press, Princeton, NJ, 1954, 167–190.Google Scholar
  26. [26]
    B. Ya. Levin, Distribution of Zeros of Entire Functions, Gostekhizdat, Moscow, 1956; English transl.: Akademie-Verlag, Berlin, 1962; Amer. Math. Soc., Providence, RI, 1964.Google Scholar
  27. [27]
    V. B. Lidskii, “Summability of series in principal vectors of non-selfadjoint operators,” Trudy Moskov. Mat. Obshch., 11 (1962), 3–35; English transl.: Amer. Math. Soc. Transl. (2), vol. 40, 1964, 193–228.MathSciNetGoogle Scholar
  28. [28]
    A. S. Markus, “Some criteria for the completeness of a system of root vectors of a linear operator in a Banach space,” Mat. Sb., 70 (112):4 (1966), 526–561.MathSciNetGoogle Scholar
  29. [29]
    A. S. Markus and V. I. Matsaev, “Analogs of Weyl inequalities in a Banach space,” Matem. Sb., 86:2 (1971), 299–313; English transl.: Math. USSR-Sb., 15 (1071), 299–312.CrossRefGoogle Scholar
  30. [30]
    V. I. Matsaev, “A method for the estimation of the resolvents of non-selfadjoint operators,” Dokl. Akad. Nauk SSSR, 154 (1964); English transl.: Soviet Math. Dokl., 5 (1964), 236–240.Google Scholar
  31. [31]
    S. Mizohata, The Theory of Partial Differential Equations, Cambridge Univ. Press, Cambridge, 1973.MATHGoogle Scholar
  32. [32]
    M. Mitrea, “The initial Dirichlet boundary value problem for general second order parabolic systems in nonsmooth manifolds,” Comm. Partial Differential Equations, 26:11–12 (2001), 1975–2036.MATHCrossRefMathSciNetGoogle Scholar
  33. [33]
    M. Mitrea, M. Taylor, “Potential theory on Lipschitz domains in Riemannian manifolds: Sobolev-Besov space results and the Poisson problem,” J. Funct. Anal., 176:1 (2000), 1–79.MATHCrossRefMathSciNetGoogle Scholar
  34. [34]
    L. Nirenberg, “Remarks on strongly elliptic partial differential equations,” Comm. Pure Appl. Math., 8 (1955), 649–675.MATHCrossRefMathSciNetGoogle Scholar
  35. [35]
    A. Pietsch, Eigenvalues and s-Numbers, Acad. Verl., Leipzig, 1987; Cambridge Studies in Adv. Math., vol. 13, Cambridge University Press, Cambridge, 1987.Google Scholar
  36. [36]
    V. S. Rychkov, “On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains,” J. London Math. Soc. (2), 60:1 (1999), 237–257.CrossRefMathSciNetGoogle Scholar
  37. [37]
    J. Savarée, “Regularity results for elliptic equations in Lipschitz domains,” J. Funct. Anal., 152:1 (1998), 176–201.CrossRefMathSciNetGoogle Scholar
  38. [38]
    Z. Shen, “Resolvent estimates in L p for elliptic systems in Lipschitz domains,” J. Funct. Anal., 133:1 (1995), 224–251.MATHCrossRefMathSciNetGoogle Scholar
  39. [39]
    I. Ya. Shneiberg, “Spectral properties of linear operators in interpolation families of Banach spaces,” Mat. Issled., 9:2 (1974), 214–227.MATHMathSciNetGoogle Scholar
  40. [40]
    T. A. Suslina, “Spectral asymptotics of variational problems with elliptic constraints in domains with piecewise smooth boundary,” Russian J. Math. Phys., 6:2 (1999), 214–234.MATHMathSciNetGoogle Scholar
  41. [41]
    H. Triebel, Interpolation, Function spaces, Differential Operators, North-Holland, Amsterdam, 1978.Google Scholar
  42. [42]
    H. Triebel, “Function spaces in Lipschitz domains and on Lipschitz manifolds. Characteristic functions as pointwise multipliers,” Rev. Mat. Comput., 15:2 (2002), 475–524.MATHMathSciNetGoogle Scholar
  43. [43]
    M. I. Vishik, “On strongly elliptic systems of differential equations,” Mat. Sb., 29 (71):3 (1951), 615–676.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Moscow Institute of Electronics and MathematicsMoscowRussia

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