Optimal regular differential operators with variable coefficients and applications
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RESEARCH
Open Access
Optimal regular differential operators with variable coefficients and applications Veli Shakhmurov* *
Correspondence: [email protected] Department of Mechanical Engineering, Okan University, Akfirat, Tuzla, Istanbul 34959, Turkey
Abstract In this paper, maximal regularity properties for linear and nonlinear high-order elliptic differential-operator equations with VMO coefficients are studied. For the linear case, the uniform coercivity property of parameter-dependent boundary value problems is obtained in Lp spaces. Then, the existence and uniqueness of a strong solution of the boundary value problem for a high-order nonlinear equation are established. In application, the maximal regularity properties of the anisotropic elliptic equation and the system of equations with VMO coefficients are derived. AMS Subject Classification: 58I10; 58I20; 35Bxx; 35Dxx; 47Hxx; 47Dxx Keywords: differential equations with VMO coefficients; boundary value problems; differential-operator equations; maximal Lp regularity; abstract function spaces; nonlinear elliptic equations
1 Introduction The goal of the present paper is to study the nonlocal boundary value problems (BVPs) for parameter-dependent linear differential-operator equations (DOEs) with discontinuous top-order coefficients sa(x)u(m) (x) + A(x)u(x) +
m–
k
s m Ak (x)u(k) (x) + λu(x) = f (x),
()
k=
and the nonlinear equation a(x)u(m) (x) + B x, u, u() , . . . , u(m–) u(x) = F x, u, u() , . . . , u(m–) , where a is a complex-valued function, s is a positive and λ is a complex parameter; A = A(x), Ak = Ak (x) are linear and B is a nonlinear operator in a Banach space E. Here the principal coefficients a and A may be discontinuous. More precisely, we assume that a and A(·)A– (x ) belong to the operator-valued Sarason class VMO (vanishing mean oscillation). Sarason class VMO was at first defined in []. In the recent years, there has been considerable interest to elliptic and parabolic equations with VMO coefficients. This ¯ that ensures the is mainly due to the fact that VMO spaces contain as a subspace C() extension of Lp -theory of operators with continuous coefficients to discontinuous coefσ ficients (see, e.g., [–]). On the other hand, the Sobolev spaces W ,n () and W σ , n (), © 2013 Shakhmurov; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Shakhmurov Fixed Point Theory and Applications 2013, 2013:42 http://www.fixedpointtheoryandapplications.com/content/2013/1/42
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< σ < , are also contained in VMO. Global regularity of the Dirichlet problem for elliptic equations with VMO coefficients has been studied in [–]. We refer to the survey [], where excellent presentation and relations with similar results can be found concerning the regularizing properties of these operators
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