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

Abstract

We study elliptic and parabolic problems governed by the singular elliptic operators $$ \mathcal L=y^{α_1}\mbox{Tr }\left(QD^2_x\right)+2y^{\frac{α_1+α_2}{2}}q\cdot \nabla_xD_y+γy^{α_2} D_{yy}+y^{\frac{α_1+α_2}{2}-1}\left(d,\nabla_x\right)+cy^{α_2-1}D_y-by^{α_2-2}$$ in the half-space ${\mathcal{R}}_{+}^{N+1}=\{(x,y):x\in {\mathcal{R}}^N,y>0\}$, under Dirichlet or oblique derivative boundary conditions. In the special case ${\alpha }_1={\alpha }_2=\alpha$ the operator $\mathcal{L}$ takes the form $$ \mathcal L=y^α\mbox{Tr }\left(AD^2\right)+y^{α-1}\left(v,\nabla\right)-by^{α-2},$$ where $v=(d,c)\in {\mathcal{R}}^{N+1}$, $b\in \mathcal{R}$ and $A=\left(\begin{array}{cc}Q& q^t\\ q& \gamma \end{array}\right)$ 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 $\mathcal{L}$ generates an analytic semigroup, characterize its domain as a weighted Sobolev space and show that it has maximal regularity.