Regularity theory for parabolic operators in the half-space with boundary degeneracy

TL;DR

This paper establishes regularity results and solvability for elliptic and parabolic problems involving boundary-degenerate elliptic operators in a half-space, demonstrating that these operators generate analytic semigroups with maximal regularity.

Contribution

It introduces new $L^p$-estimates and characterizes the domain of boundary-degenerate operators, advancing the understanding of their analytic semigroup generation and regularity properties.

Findings

Proves elliptic and parabolic $L^p$-estimates for degenerate operators.

Shows the operator generates an analytic semigroup.

Characterizes the domain as a weighted Sobolev space.

Abstract

We study elliptic and parabolic problems governed by the singular elliptic operators \begin{align*} \mathcal L=y^{\alpha_1}\mbox{Tr }\left(QD^2_xu\right)+2y^{\frac{\alpha_1+\alpha_2}{2}}q\cdot \nabla_xD_y+\gamma y^{\alpha_2} D_{yy}+Cy^{\alpha_2-1}D_y \end{align*} under Neumann boundary condition, in the half-space $\mathbb{R}^{N+1}_+=\{(x,y): x \in \mathbb{R}^N, y>0\}$. 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.