A unified approach to degenerate problems in the half-space

TL;DR

This paper develops a unified mathematical framework to analyze elliptic and parabolic problems involving singular elliptic operators in the half-space, addressing degeneracies caused by variable coefficients and singularities.

Contribution

It introduces a comprehensive approach to handle degenerate elliptic and parabolic operators with singular coefficients in the half-space setting.

Findings

Established well-posedness of problems with singular elliptic operators.

Derived regularity results for solutions near degeneracy points.

Provided a unified method applicable to various degenerate PDEs.

Abstract

We study elliptic and parabolic problems governed by the singular elliptic operators \begin{equation*} \mathcal L =y^{\alpha_1}\Delta_{x} +y^{\alpha_2}\left(D_{yy}+\frac{c}{y}D_y -\frac{b}{y^2}\right), \qquad\alpha_1, \alpha_2 \in\mathbb R \end{equation*} in the half-space $\mathbb R^{N+1}_+=\{(x,y): x \in \mathbb R^N, y>0\}$.