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.
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References
Sh. Agmon, “On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems,” Comm. Pure Appl. Math., 15:2 (1962), 119–147.
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.
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.
M. S. Agranovich, “On a mixed Poincaré-Steklov type spectral problem in a Lipschitz domain,” Russian J. Math. Phys., 13:3 (2006), 239–244.
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.
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.
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.
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.
J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-New York, 1976.
M. S. Birman and M. Z. Solomyak, “Piecewise-polynomial approximations of functions of the classes W α p ,” Mat. Sb., 73:3 (1967), 331–355.
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.
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.
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.
J. Burgoyne, “Denseness of the generalized eigenvectors of a discrete operator in a Banach space,” J. Operator Theory, 33 (1995), 279–297.
N. Dunford and J. T. Schwartz, Linear Operators, Part II, Interscience Publishers, New York-London, 1963.
D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford Univ. Press, Oxford, 1987.
D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, and Differential Operators, Cambridge Univ. Press, Cambridge, 1996.
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.
A. Grothendieck, “Produits tenzoriels topologiques et espaces nucléares,” Mem. Amer. Math. Soc., 16 (1955).
A. Grothendieck, “La théorie de Fredholm,” Bull. Soc. Math. France, 84 (1956), 319–384.
S. Janson, P. Nilsson, and J. Peetre, “Notes on Wolff’s note on interpolation spaces,” Proc. London Math. Soc., 48:2 (1984), 283–299.
A. Jonsson and H. Wallin, Function Spaces on Subsets of ℝn, Mathematical Reports, vol. 2, part 1, Harwood Academic Publishers, London etc., 1984.
H. König, Eigenvalue Distribution of Compact Operators, Operator Theory: Advances and Applications, vol. 16, Birkhäuser, Basel etc., 1986.
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.
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.
B. Ya. Levin, Distribution of Zeros of Entire Functions, Gostekhizdat, Moscow, 1956; English transl.: Akademie-Verlag, Berlin, 1962; Amer. Math. Soc., Providence, RI, 1964.
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.
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.
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.
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.
S. Mizohata, The Theory of Partial Differential Equations, Cambridge Univ. Press, Cambridge, 1973.
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.
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.
L. Nirenberg, “Remarks on strongly elliptic partial differential equations,” Comm. Pure Appl. Math., 8 (1955), 649–675.
A. Pietsch, Eigenvalues and s-Numbers, Acad. Verl., Leipzig, 1987; Cambridge Studies in Adv. Math., vol. 13, Cambridge University Press, Cambridge, 1987.
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.
J. Savarée, “Regularity results for elliptic equations in Lipschitz domains,” J. Funct. Anal., 152:1 (1998), 176–201.
Z. Shen, “Resolvent estimates in L p for elliptic systems in Lipschitz domains,” J. Funct. Anal., 133:1 (1995), 224–251.
I. Ya. Shneiberg, “Spectral properties of linear operators in interpolation families of Banach spaces,” Mat. Issled., 9:2 (1974), 214–227.
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.
H. Triebel, Interpolation, Function spaces, Differential Operators, North-Holland, Amsterdam, 1978.
H. Triebel, “Function spaces in Lipschitz domains and on Lipschitz manifolds. Characteristic functions as pointwise multipliers,” Rev. Mat. Comput., 15:2 (2002), 475–524.
M. I. Vishik, “On strongly elliptic systems of differential equations,” Mat. Sb., 29 (71):3 (1951), 615–676.
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To dear Israel Moiseevich Gelfand in connection with his 95th birthday
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Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 42, No. 4, pp. 2–23, 2008
Original Russian Text Copyright © by M. S. Agranovich
Supported by RFBR grant no. 07-01-00287.
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Agranovich, M.S. Spectral boundary value problems in Lipschitz domains for strongly elliptic systems in Banach spaces H σ p and B σ p . Funct Anal Its Appl 42, 249–267 (2008). https://doi.org/10.1007/s10688-008-0039-x
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DOI: https://doi.org/10.1007/s10688-008-0039-x