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

TL;DR

This paper investigates singular elliptic and parabolic operators in the half-space with boundary conditions, establishing $L^p$-estimates, solvability, and semigroup properties for these degenerate operators.

Contribution

It provides new $L^p$-estimates, solvability results, and semigroup characterizations for singular parabolic operators with boundary degeneracy, extending existing theory.

Findings

Proved elliptic and parabolic $L^p$-estimates and solvability.

Showed the operator generates an analytic semigroup.

Characterized the domain as a weighted Sobolev space.

Abstract

We study elliptic and parabolic problems governed by the singular elliptic operators $$ \mathcal L=y^{\alpha_1}\mbox{Tr }\left(QD^2_x\right)+2y^{\frac{\alpha_1+\alpha_2}{2}}q\cdot \nabla_xD_y+\gamma y^{\alpha_2} D_{yy}+y^{\frac{\alpha_1+\alpha_2}{2}-1}\left(d,\nabla_x\right)+cy^{\alpha_2-1}D_y-by^{\alpha_2-2}$$ 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 $$ \mathcal L=y^{\alpha}\mbox{Tr }\left(AD^2\right)+y^{\alpha-1}\left(v,\nabla\right)-by^{\alpha-2},$$ where $v=(d,c)\in\mathcal{R}^{N+1}$, $b\in\mathcal{R}$ and $ A=\left( \begin{array}{c|c} Q & { q}^t \\[1ex] \hline 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.