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Regular degenerate separable differential operators and applications. (English) Zbl 1229.34093
Summary: Consider on $(0,1)$ the boundary value problem $$\aligned & Lu=-a(x)u^{[2]}(x)+A(x)u(x)+A_1(x)u^{[1]}(x)+A_2(x)u(x)=f,\\ & L_1u=\sum^{m_1}_{k=0}\alpha_ku^{[k]}(0)=0,\quad L_2u=\sum^{m_2}_{k=0}\beta_ku^{[k]}(1)=0\endaligned\tag*$$ in $L_p(0,1;E)$, where $u^{[i]}=\left[x^{\gamma_1}(1-x)^{\gamma_2}\frac{d}{dx}\right]^iu(x)$, $0\le\gamma_i<1$, $m_k\in\{0,1\}$; $\alpha_k$ and $\beta_k$ are complex numbers, $A$ and $A_i(x)$ are linear operators in a Banach space $E$. Several conditions for separability, Fredholmness and resolvent estimates in $L_p$-spaces are given. As applications, the degenerate Cauchy problem for parabolic equations, boundary value problems for degenerate partial differential equations and systems of degenerate elliptic equations on a cylindrical domain are studied.

34G10Linear ODE in abstract spaces
35J25Second order elliptic equations, boundary value problems
35J70Degenerate elliptic equations
35K65Parabolic equations of degenerate type
34B15Nonlinear boundary value problems for ODE
Full Text: DOI
[1] Agmon, S.: On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems. Commun. Pure Appl. Math. 15, 119--147 (1962) · Zbl 0109.32701 · doi:10.1002/cpa.3160150203
[2] Ashyralyev, A.: On well-posedeness of the nonlocal boundary value problem for elliptic equations. Numer. Funct. Anal. Optim. 24(1 &amp; 2), 1--15 (2003) · Zbl 1055.35018 · doi:10.1081/NFA-120020240
[3] Amann, H.: Linear and Quasi-Linear Equations,1. Birkhauser, Basel (1995) · Zbl 0819.35001
[4] Aubin, J.P.: Abstract boundary-value operators and their adjoint. Rend. Semin. Mat. Univ. Padova 43, 1--33 (1970) · Zbl 0247.47042
[5] Agarwal, R., O’ Regan, D., Shakhmurov, V.B.: Separable anisotropic differential operators in weighted abstract spaces and applications. J. Math. Anal. Appl. 338, 970--983 (2008) · Zbl 1138.47037 · doi:10.1016/j.jmaa.2007.05.078
[6] Besov, O.V., Ilin, V.P., Nikolskii, S.M.: Integral Representations of Functions and Embedding Theorems. Nauka, Moscow (1975)
[7] Burkholder, D.L.: A geometrical conditions that implies the existence certain singular integral of Banach space-valued functions. In: Proc. Conf. Harmonic Analysis in Honor of Antonu Zigmund, Chicago, 1981, pp. 270--286. Wads Worth, Belmont (1983)
[8] Clement, Ph., de Pagter, B., Sukochev, F.A., Witvliet, H.: Schauder decomposition and multiplier theorems. Stud. Math. 138, 135--163 (2000) · Zbl 0955.46004
[9] Dore, C., Yakubov, S.: Semigroup estimates and non coercive boundary value problems. Semigroup Forum 60, 93--121 (2000) · Zbl 0973.20055 · doi:10.1007/s002330010005
[10] Dore, G., Venni, A.: On the closedness of the sum of two closed operators. Math. Z. 196, 189--201 (1987) · Zbl 0615.47002 · doi:10.1007/BF01163654
[11] Denk, R., Hieber, M., Prüss, J.: R-boundedness, fourier multipliers and problems of elliptic and parabolic type. Mem. Am. Math. Soc. 166(788), Providence RI (2003) · Zbl 1274.35002
[12] Favini, A., Shakhmurov, V.B., Yakubov, Y.: Regular boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces. Semigroup form. 79(1), 22--54 (2009) · Zbl 1177.47089 · doi:10.1007/s00233-009-9138-0
[13] Grisvard, P.: Commutative’ de deux foncteurs d’interpolation et applications. J. Math. Pures Appl. 45(9), 143--290 (1966) · Zbl 0173.15803
[14] Haller, R., Heck, H., Noll, A.: Mikhlin’s theorem for operator-valued Fourier multipliers in n variables. Math. Nachr. 244, 110--130 (2002) · Zbl 1054.47013 · doi:10.1002/1522-2616(200210)244:1<110::AID-MANA110>3.0.CO;2-S
[15] Goldstain, J.A.: Semigroups of Linear Operators and Applications. Oxford Mathematical Monographs. Oxford University Press, Clarendon Press, New York and Oxford (1985)
[16] Krein, S.G.: Linear Differential Equations in Banach Space. American Mathematical Society, Providence (1971) · Zbl 0236.47034
[17] Komatsu, H.: Fractional powers of operators. Pac. J. Math. 19, 285--346 (1966) · Zbl 0154.16104 · doi:10.2140/pjm.1966.19.285
[18] Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhauser (2003) · Zbl 0816.35001
[19] Lizorkin, P.I.: $$\backslash$left( L_{p},L_{q}$\backslash$right) $ -multiplicators of fourier integrals. Dokl. Akad. Nauk SSSR 152(4), 808--811 (1963)
[20] Lamberton, D.: Equations d’evolution linéaires associées à des semi-groupes de contractions dans les espaces L p . J. Funct. Anal. 72, 252--262 (1987) · Zbl 0621.47039 · doi:10.1016/0022-1236(87)90088-7
[21] Nazarov, S.A., Plammenevskii, B.A.: Elliptic Problems in Domains with Piecewise Smooth Boundaries. Walter de Gruyter, New York (1994)
[22] Shklyar, A.Ya.: Complete Second Order Linear Differential Equations in Hilbert Spaces. Birkhauser Verlak, Basel (1997) · Zbl 0873.34049
[23] Sobolevskii, P.E.: Coerciveness inequalities for abstract parabolic equations. Dokl. Akad. Nauk SSSR 57(1), 27--40 (1964)
[24] Shakhmurov, V.B.: Theorems about of compact embedding and applications. Dokl. Akad. Nauk SSSR 241(6), 1285--1288 (1978)
[25] Shakhmurov, V.B.: Theorems on compactness of embedding in weighted anisotropic spaces, and their applications. Dokl. Akad. Nauk SSSR 291(6), 612--616 (1986) · Zbl 0638.46027
[26] Shakhmurov, V.B.: Imbedding theorems and their applications to degenerate equations. Diff. Equ. 24(4), 475--482 (1988) · Zbl 0672.46014
[27] Shakhmurov, V.B.: Coercive boundary value problems for regular degenerate differential-operator equations. J. Math. Anal. Appl. 292(2), 605--620 (2004) · Zbl 1060.35045 · doi:10.1016/j.jmaa.2003.12.032
[28] Shakhmurov, V.B.: Embedding theorems and maximal regular differential operator equations in Banach-valued function spaces. J. Inequal. Appl. 2(4), 329--345 (2005) · Zbl 1119.46034
[29] Shakhmurov, V.B.: Separable anisotropic differential operators and applications. J. Math. Anal. Appl. 327(2), 1182--1201 (2006) · Zbl 1113.47035 · doi:10.1016/j.jmaa.2006.05.007
[30] Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978) · Zbl 0387.46032
[31] Weis, L.: Operator-valued Fourier multiplier theorems and maximal L p regularity. Math. Ann. 319, 735--758 (2001) · Zbl 0989.47025 · doi:10.1007/PL00004457
[32] Yakubov, S.: Completeness of Root Functions of Regular Differential Operators. Longman, Scientific and Technical, New York (1994) · Zbl 0833.34081
[33] Yakubov, S.: A nonlocal boundary value problem for elliptic differential-operator equations and applications. Integr. Equ. Oper. Theory 35, 485--506 (1999) · Zbl 0942.35061 · doi:10.1007/BF01228044
[34] Yakubov, S., Yakubov, Ya.: Differential-Operator Equations. Ordinary and Partial Differential Equations. Chapman and Hall/CRC, Boca Raton (2000) · Zbl 0936.35002