Riv. Mat. Univ. Parma, Vol. 5, No. 2, 2014

Angelo Favini[1] and Yakov Yakubov[2]

Isomorphism for regular boundary value problems for elliptic differential-operator equations of the fourth order depending on a parameter

Pages: 335-361
Accepted in revised form: 15 October 2013
Mathematics Subject Classification (2010): 34G10, 34L10, 35J40, 35P10, 47E05, 47N20.

Keywords: Abstract elliptic equation, quasi-elliptic equations, UMD Banach space, isomorphism, completeness of eigenfunctions, maximal $$L_p$$-regularity.
[1] : Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy
[2] : Raymond and Beverly Sackler, School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel

Abstract: We treat some fourth order elliptic differential-operator boundary value problems on a finite interval quadratically depending on a parameter. We prove an isomorphism result (which implies maximal $$L_p$$-regularity) in the corresponding abstract Sobolev spaces. The underlying space is a UMD Banach space. Then, for the corresponding homogeneous problems, we prove discreteness of the spectrum and two-fold completeness of a system of eigenvectors and associated vectors of the problem in the framework of Hilbert and UMD Banach spaces. We apply the obtained abstract results to non-local boundary value problems for elliptic and quasi-elliptic equations with a parameter in (bounded and unbounded) cylindrical domains.

The first author is a member of G.N.A.M.P.A. and the paper fits the RFO research program of Ministero della Pubblica Istruzione, Università e Ricerca; the second author was supported by RFO 2010 funds and by the Israel Ministry of Absortion.

References

[1] B. A. Aliev and Ya. Yakubov, Elliptic differential-operator problems with a spectral parameter in both the equation and boundary-operator conditions, Adv. Differential Equations 11 (2006), no. 10, 1081-1110. [MR2279710] (Erratum: ibid 12 (2007), no. 9, 1079. [MR2351838])
[2] B. A. Aliev and Ya. Yakubov, Second order elliptic differential-operator equations with unbounded operator boundary conditions in UMD Banach spaces, Integral Equations Operator Theory 69 (2011), 269-300. [MR2765589]
[3] R. Denk, M. Hieber and J. Pruss, $$\mathcal R$$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc. 166 (2003), no. 788. [MR2006641]
[4] A. Favini, D. Guidetti and Ya. Yakubov, Abstract elliptic and parabolic systems with applications to problems in cylindrical domains, Adv. Differential Equations 16 (2011), no. 11-12, 1139-1196. [MR2858526]
[5] A. Favini, V. Shakhmurov and Ya. Yakubov, Regular boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces, Semigroup Forum 79 (2009), 22-54. [MR2534222]
[6] A. Favini and Ya. Yakubov, Higher order ordinary differential-operator equations on the whole axis in UMD Banach spaces, Differential Integral Equations 21 (2008), no. 5-6, 497-512. [MR2483266]
[7] A. Favini and Ya. Yakubov, Regular boundary value problems for elliptic differential-operator equations of the fourth order in UMD Banach spaces, Sci. Math. Jpn. 70 (2009), 183-204. [MR2555735]
[8] A. Favini and Ya. Yakubov, Irregular boundary value problems for second order elliptic differential-operator equations in UMD Banach spaces, Math. Ann. 348 (2010), 601-632. [MR2677897]
[9] A. Favini and Ya. Yakubov, Regular boundary value problems for ordinary differential-operator equations of higher order in UMD Banach spaces, Discrete Contin. Dyn. Syst. Ser. S 4 (2011), no. 3, 595-614. [MR2746422]
[10] N. Kalton, P. Kunstmann and L. Weis, Perturbation and interpolation theorems for the $$H^{\infty}$$-calculus with applications to differential operators, Math. Ann. 336 (2006), 747-801. [MR2255174]
[11] R. S. Phillips, On weakly compact subsets of a Banach space, Amer. J. Math. 65 (1943), 108-136. [MR0007938]
[12] H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland Publishing Co., Amsterdam-New York 1978. [MR0503903]
[13] S. Yakubov and Ya. Yakubov, Differential-operator equations. Ordinary and partial differential equations, Chapman & Hall/CRC, Boca Raton, FL 2000. [MR1739280]
[14] Ya. Yakubov, Fold completeness of a system of root vectors of a system of unbounded polynomial operator pencils in Banach spaces. I. Abstract theory, J. Math. Pures Appl. 92 (2009), no. 3, 263-275. [MR2555179]
[15] F. Zimmermann, On vector-valued Fourier multiplier theorems, Studia Math. 93 (1989), no. 3, 201-222. [MR1030488]

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