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=y^₁Tr (QD²ₓ) +2y^₁+₂{2}q ₓDᵧ+ y^₂ Dₘₘ+y^₁+₂{2-1} (d, ₓ) +cy^₂-1Dᵧ-by^₂-2 in the half-space R^N+1_+=\ (x, y): x RN, y>0\, under Dirichlet or oblique derivative boundary conditions. In the special case ₁=₂= the operator L takes the form L=y^Tr (AD²) +y^-1 (v, ) -by^-2, where v= (d, c) ^N+1, b and A= (arrayc|c Q & qᵗ \\1ex q& array) is an elliptic matrix. We prove elliptic and parabolic Lᵖ-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.