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

We consider the two-dimensional differential operator Au(x(1),x(2)) = -a(11)(x(1),x(2))u(x1x1) (x(1),x(2)) - a(22)(x(1),x(2))u(x2x2) (x(1),x(2)) + sigma u(x(1),x(2)) defined on functions on the half-plane Omega = R+ x R with the boundary conditions u(0,x(2)) = 0, x(2) is an element of R, where a(ii)(x(1),x(2)), i = 1, 2, are continuously differentiable and satisfy the uniform ellipticity condition a(11)(2)(x(1),x(2)) + a(22)(2)(x(1),x(2)) >= delta > 0, sigma > 0. The structure of the fractional spaces E-alpha(A,C-beta(Omega)) generated by the operator A is investigated. The positivity of A in Holder spaces is established. In applications, theorems on well-posedness in a Holder space of elliptic problems are obtained.