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
Received: 20 March 2013   
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.
Authors addresses:
[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.

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