Singular parabolic problems in the half-space

We study elliptic and parabolic problems governed by singular elliptic operators $$\mathcal L =\sum _{i,j=1}^{N+1}q_{ij}D_{ij}+\frac c y D_y$$ in the half-space $\mathbb R ^{N+1}_+=\{(x,y): x \in \mathbb R ^N$, $y \gt 0\}$ under Neumann boundary conditions at $y=0$. More general operators and oblique derivative boundary conditions are also considered.