- Research Article
- Open access
- Published:
Embedding theorems in Banach-valued
-spaces and maximal
-regular differential-operator equations
Journal of Inequalities and Applications volume 2006, Article number: 16192 (2006)
Abstract
The embedding theorems in anisotropic Besov-Lions type spaces are studied; here
and
are two Banach spaces. The most regular spaces
are found such that the mixed differential operators
are bounded from
to
, where
are interpolation spaces between
and
depending on
and
. By using these results the separability of anisotropic differential-operator equations with dependent coefficients in principal part and the maximal
-regularity of parabolic Cauchy problem are obtained. In applications, the infinite systems of the quasielliptic partial differential equations and the parabolic Cauchy problems are studied.
References
Agmon S, Nirenberg L: Properties of solutions of ordinary differential equations in Banach space. Communications on Pure and Applied Mathematics 1963, 16: 121–239. 10.1002/cpa.3160160204
Agranovič MS, Višik MI: Elliptic problems with a parameter and parabolic problems of general type. Uspekhi Matematicheskikh Nauk 1964,19(3 (117)):53–161.
Amann H: Linear and Quasilinear Parabolic Problems. Vol. I, Monographs in Mathematics. Volume 89. Birkhäuser Boston, Massachusetts; 1995:xxxvi+335.
Amann H: Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications. Mathematische Nachrichten 1997, 186: 5–56.
Amann H: Compact embeddings of vector-valued Sobolev and Besov spaces. Glasnik Matematički. Serija III 2000,35(55)(1):161–177.
Ashyralyev A: On well-posedness of the nonlocal boundary value problems for elliptic equations. Numerical Functional Analysis and Optimization 2003,24(1–2):1–15. 10.1081/NFA-120020240
Aubin J-P: Abstract boundary-value operators and their adjoints. Rendiconti del Seminario Matematico della Università di Padova 1970, 43: 1–33.
Besov OV, Il'in VP, Nikol'skiĭ SM: Integral representations of functions, and embedding theorems. Izdat. "Nauka", Moscow; 1975:480.
Burkholder DL: A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions. In Proceedings of Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser.. Wadsworth, California; 1983:270–286.
Calderón A-P: Intermediate spaces and interpolation, the complex method. Studia Mathematica 1964, 24: 113–190.
Clément P, de Pagter B, Sukochev FA, Witvliet H: Schauder decomposition and multiplier theorems. Studia Mathematica 2000,138(2):135–163.
Denk R, Hieber M, Prüss J: -boundedness, Fourier multipliers and problems of elliptic and parabolic type. Memoirs of the American Mathematical Society 2003,166(788):viii+114.
Dore G, Yakubov S: Semigroup estimates and noncoercive boundary value problems. Semigroup Forum 2000,60(1):93–121. 10.1007/s002330010005
Girardi M, Weis L: Operator-valued Fourier multiplier theorems on Besov spaces. Mathematische Nachrichten 2003,251(1):34–51. 10.1002/mana.200310029
Gorbachuk VI, Gorbachuk ML: Boundary value problems for operator-differential equations. "Naukova Dumka", Kiev; 1984:284.
Grisvard P: Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics. Volume 24. Pitman, Massachusetts; 1985:xiv+410.
Komatsu H: Fractional powers of operators. Pacific Journal of Mathematics 1966,19(2):285–346.
Kreĭn SG: Linear Differential Equations in Banach Space. American Mathematical Society, Rhode Island; 1971:v+390.
Lindenstrauss J, Tzafriri L: Classical Banach Spaces. II. Function Spaces, Results in Mathematics and Related Areas. Volume 97. Springer, New York; 1979:x+243.
Lions J-L, Magenes E: Problèmes aux limites non homogénes. VI [Problems and limites non homogenes]. Journal d'Analyse Mathématique 1963, 11: 165–188.
Lions J-L, Peetre J: Sur une classe d'espaces d'interpolation. Institut des Hautes Études Scientifiques. Publications Mathématiques 1964, 19: 5–68. 10.1007/BF02684796
Lizorkin PI: -multipliers of Fourier integrals. Doklady Akademii Nauk SSSR 1963,152(4):808–811.
Lizorkin PI, Shakhmurov VB: Embedding theorems for vector-valued functions. I. Izvestiya Vysshikh Uchebnykh Zavedeniĭ. Matematika 1989, (1):70–79.
Lizorkin PI, Shakhmurov VB: Embedding theorems for vector-valued functions. II. Izvestiya Vysshikh Uchebnykh Zavedeniĭ. Matematika 1989, (2):47–54.
McConnell TR: On Fourier multiplier transformations of Banach-valued functions. Transactions of the American Mathematical Society 1984,285(2):739–757. 10.1090/S0002-9947-1984-0752501-X
Nazarov SA, Plamenevsky BA: Elliptic Problems in Domains with Piecewise Smooth Boundaries, de Gruyter Expositions in Mathematics. Volume 13. Walter de Gruyter, Berlin; 1994:viii+525.
Schmeisser H-J: Vector-valued Sobolev and Besov spaces. In Seminar Analysis of the Karl-Weierstraß-Institute of Mathematics 1985/1986 (Berlin, 1985/1986), Teubner-Texte Math.. Volume 96. Teubner, Leipzig; 1987:4–44.
Shakhmurov VB: Imbedding theorems for abstract function- spaces and their applications. Mathematics of the USSR-Sbornik 1987,134(1–2):261–276.
Shakhmurov VB: Embedding theorems and their applications to degenerate equations. Differential Equations 1988,24(4):475–482.
Shakhmurov VB: Coercive boundary value problems for regular degenerate differential-operator equations. Journal of Mathematical Analysis and Applications 2004,292(2):605–620. 10.1016/j.jmaa.2003.12.032
Shakhmurov VB: Embedding operators and maximal regular differential-operator equations in Banach-valued function spaces. Journal of Inequalities and Applications 2005,2005(4):329–345. 10.1155/JIA.2005.329
Shakhmurov VB, Dzhabrailov MS: On the compactness of the embedding of-spaces. Akademiya Nauk Azerbaĭ dzhanskoĭ SSR. Doklady, TXLVI 1990,46(3):7–10 (1992).
Shklyar AYa: Complete Second Order Linear Differential Equations in Hilbert Spaces, Operator Theory: Advances and Applications. Volume 92. Birkhäuser, Basel; 1997:xii+219.
Sobolev SL: Imbedding theorems for abstract functions of sets. Doklady Akademii Nauk SSSR 1957, 115: 57–59.
Sobolev SL: Certain Applications of Functional Analysis to Mathematical Physics. Nauka, Moscow; 1968.
Sobolevskiĭ PE: Coerciveness inequalities for abstract parabolic equations. Doklady Akademii Nauk SSSR 1964, 157: 52–55.
Triebel H: Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library. Volume 18. North-Holland, Amsterdam; 1978:528.
Triebel H: Fractals and Spectra, Monographs in Mathematics. Volume 91. Birkhäuser, Basel; 1997:viii+271.
Yakubov S: A nonlocal boundary value problem for elliptic differential-operator equations and applications. Integral Equations and Operator Theory 1999,35(4):485–506. 10.1007/BF01228044
Yakubov S, Yakubov Ya: Differential-Operator Equations. Ordinary and Partial Differential Equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics. Volume 103. Chapman & Hall/CRC, Florida; 2000:xxvi+541.
Zimmermann F: On vector-valued Fourier multiplier theorems. Studia Mathematica 1989,93(3):201–222.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Shakhmurov, V.B. Embedding theorems in Banach-valued-spaces and maximal
-regular differential-operator equations.
J Inequal Appl 2006, 16192 (2006). https://doi.org/10.1155/JIA/2006/16192
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/JIA/2006/16192