The structure of fractional spaces generated by the two-dimensional difference operator on the half plane

Abstract

We consider a difference operator approximation $A^x_h$ of the differential operator $A^xu(x) = a_{11}(x)u_{x_1 x_1}(x) - a_{22}(x)u_{x_2x_2} (x) + \sigma u(x),\; x = (x_1, x_2)$ defined in the region $R^{+} \times R$ with the boundary condition $u(0, x_2) = 0,\; x_2 \in R$. Here, the coefficients $a_{ii}(x), i = 1, 2$, are continuously differentiable, satisfy the uniform ellipticity condition $a^2_{11}(x) + a^2_{22}(x) \geq \delta > 0$. We investigate the structure of the fractional spaces generated by the analyzed difference operator. Theorems on well-posedness in a Holder space of difference elliptic problems are obtained as applications.