A note on positivity of two-dimensional differential operators

Abstract

We consider the two-dimensional differential operator A(t,x)u(t,x) = -a11 (t, x) utt -a22(t,x)uxx +?u defined on functions on the half-plane R+ x R with the boundary condition u(0,x) = 0, x ? R where aii(t,x), i = 1,2 are continuously differentiable and satisfy the uniform ellipticity condition a2 11(t,x) + a222(t,x)? ? > 0, ? > 0. The structure of fractional spaces E?,1 (L1 (R+ x R), A(t,x)) generated by the operator A(t,x) is investigated. The positivity of A(t,x) in L1 (W2?1(R+ x R)) spaces is established. In applications, theorems on well-posedness in L1 (W2?1 (R+ x R)) spaces of elliptic problems are obtained.