A note on positivity of two-dimensional differential operators

We consider the two-dimensional differential operator A(t,x)u(t, x) = -a11 (t, x) utt - a22(t, x)uxx +σu defined on functions on the half-plane ℝ+ × ℝ with the boundary condition u(0, x) = 0, x ε ℝ where aii(t, x), i = 1, 2 are continuously differentiable and satisfy the uniform ellipticity condition a211(t, x) + a222(t, x) ≥ δ >,σ > 0: The structure of fractional spaces Eα,1 (L1 (ℝ+ × ℝ), A(t,x)) generated by the operator A(t,x) is investigated. The positivity of A(t,x) in L1 (W2α1 (ℝ+ × ℝ)) spaces is established. In applications, theorems on well-posedness in L1 (W2α1 (ℝ+ × ℝ)) spaces of elliptic problems are obtained. © 2017, University of Nis. All rights reserved.