Separability properties of agranovich-vishik type elliptic operators in banach space valued function classes

The main objective of the study is to discuss the BVP for an elliptic equation depending on a complex parameter, and defined on three different vector spaces: Euclidian space R_+^n , half-space R^n_+; and a bounded region Ω є R^n with a smooth (n-1) dimensional boundary Γ є R^(n-1), in E-valued function spaces, respectively. In other words, the purpose of this thesis is to discuss the separability properties (maximal regularity properties) of the general elliptic problem depending on a complex parameter in a bounded region Ω є R^n with a smooth (n-1) dimensional boundary. We consider a linear differential equation of arbitrary order with complex valued coefficients depending smoothly on the variable x and on the polynomially dependent complex parameter q. Let A(x,D,q)u = f (x) be a given elliptic equation depending on a complex parameter in the domain Ω. On the boundary G we are given general boundary conditions B_j(x_0,D,q)u_j|_Γ = g_j(x_0), containing a parameter q, that we will give in detail in the next section. Our main aim is to show the following estimate: ‖u‖_l+|q|^l ‖u‖_0 ≤c{‖f‖_(l-2m)+|q|^(l-2m) ‖f‖_0+∑_(j=1)^m ‖g_j‖_(l-m_j-1/p)^'+|q|^(l-m_j-1/p) ‖g_j‖_0^'}≤ C{‖u‖_l+|q|^l ‖u‖_0 }. The maximal regularity properties of elliptic problem in the Hilbert valued Sobolev space W^(l,2)(Ω;H) has been investigated before. Now, we generalize this to the Banach space valued Sobolev class W^(l,p)(Ω;E), where E is an UMD space. Firstly, we determine the maximal regularity properties of elliptic problems with complex parameter in the Banach space-valued Sobolev class W^(l,p) in the whole space and half space. Further, we analyze the separability properties of the parameter dependent general elliptic problem in a bounded domain with smooth boundary. This property is analyzed with an inequality which is called the coercive estimate. We encounter some limit when proving these theorems and in light of these we give some conditions that restrict us which one calls as Condition I and Condition II. So that there is no confusion, we give them the same name in every chapter.