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The structure of fractional spaces generated by a two-dimensional elliptic differential operator and its applications

Research paper by Allaberen Ashyralyev, Sema Akturk, Yasar Sozen

Indexed on: 02 Jan '14Published on: 02 Jan '14Published in: Boundary Value Problems



Abstract

We consider the two-dimensional differential operator Au(x1,x2)=−a11(x1,x2)ux1x1(x1,x2)−a22(x1,x2)ux2x2(x1,x2)+σu(x1,x2) defined on functions on the half-plane Ω=R+×R with the boundary conditions u(0,x2)=0, x2∈R, where aii(x1,x2), i=1,2, are continuously differentiable and satisfy the uniform ellipticity condition a112(x1,x2)+a222(x1,x2)≥δ>0, σ>0. The structure of the fractional spaces Eα(A,Cβ(Ω)) generated by the operator A is investigated. The positivity of A in Hölder spaces is established. In applications, theorems on well-posedness in a Hölder space of elliptic problems are obtained.MSC: 35J25, 47E05, 34B27.