Singular parabolic operators in the half-space with boundary degeneracy: Dirichlet and oblique derivative boundary conditions

We study elliptic and parabolic problems governed by the singular elliptic operators $$ L=yα_1Tr (QD^2_x)+2yα_1+α_2/2q· ∇_xD_y+γ yα_2 Dyy+yα_1+α_2/2-1(d,∇_x)+cyα_2-1D_y-byα_2-2$$ in the half-space $RN+1_+=\{(x,y): x ∈ R^N, y>0\}$, under Dirichlet or oblique derivative boundary conditions. In the special case $α_1=α_2=α$ the operator $ L$ takes the form $$ L=yαTr (AD^2)+yα-1(v,∇)-byα-2,$$ where $v=(d,c)N+1$, $b$ and $ A=( {array}{c|c} Q & { q}^t \\[1ex] q& γ {array})$ is an elliptic matrix. We prove elliptic and parabolic $L^p$-estimates and solvability for the associated problems. In the language of semigroup theory, we prove that $ L$ generates an analytic semigroup, characterize its domain as a weighted Sobolev space and show that it has maximal regularity.