Positivity of Two-dimensional Elliptic Differential Operators in Hölder Spaces

Subject Applied Mathematics
Title Positivity of Two-dimensional Elliptic Differential Operators in Hölder Spaces
Author(s) Allaberen Ashyralyev, Sema Aktürk, Yaşar Sözen
Keywords Positive operator, fractional spaces, Green’s function, Hölder spaces
Abstract

This paper considers the following operator

Au(t,x)=-a₁₁(t,x)u_{tt}(t,x)-a₂₂(t,x)u_{xx}(t,x)+σu(t,x),

defined over the region R⁺×R with the boundary condition u(0,x)=0, x∈R. Here, the coefficients a_{ii}(t,x), i=1,2 are continuously differentiable and satisfy the uniform ellipticity

a₁₁²(t,x)+a₂₂²(t,x)≥δ>0,

and σ>0. It investigates the structure of the fractional spaces generated by this operator. Moreover, the positivity of the operator in Hölder spaces is proved.