A Note on Positivity of Two-Dimensional Differential Operators

Abstract

We consider the two-dimensional differential operator $A^{(t,x)}u(t,x)=-a_{11}%
\left( t,x\right) u_{tt}-a_{22}(t,x)u_{xx}+\sigma u$ defined on functions on
the half-plane $\mathbb{R}^{+}\times \mathbb{R}$ with the boundary
condition $u(0,x)=0,$ $x\in \mathbb{R}$ where $a_{ii}(t,x),$ $i=1,2$ are
continuously differentiable and satisfy the uniform ellipticity condition $%
a_{11}^{2}(t,x)+a_{22}^{2}(t,x)\geq \delta >0,$ $\sigma >0.$ The structure
of fractional spaces $E_{\alpha ,1}\left( L_{1}\left( \mathbb{R}^{+}\times
\mathbb{R}\right) ,A^{(t,x)}\right) $ generated by the operator $A^{(t,x)}$ is
investigated. The positivity of $A^{(t,x)}$ in $L_{1}\left( W_{1}^{2\alpha }(\mathbb{R}^{+}\times
\mathbb{R})\right) $ spaces is established.
In applications, theorems on well-posedness in $L_{1}\left( W_{1}^{2\alpha }(\mathbb{R}^{+}\times
\mathbb{R})\right) $ spaces of
elliptic problems are obtained.