
Journal of Lie Theory 23 (2013), No. 3, 803825 Copyright Heldermann Verlag 2013 Interior Regularity for Degenerate Elliptic Equations with Drift on Homogeneous Groups Xiaojing Feng Dept. of Applied Mathematics, Northwestern Polytechnical University, Xi'an  Shaanxi 710129, P. R. China fxj467@mail.nwpu.edu.cn Pengcheng Niu Dept. of Applied Mathematics, Northwestern Polytechnical University, Xi'an  Shaanxi 710129, P. R. China pengchengniu@nwpu.edu.cn [Abstractpdf] Let $G$ be a homogeneous group and let $X_0$, $X_1$ , $X_2,\dots,X_{p_0}$ be left invariant real vector fields on $G$ satisfying H\"{o}rmander's rank condition. Assume that $X_1$, $X_2,\dots,X_{p_0}$ are homogeneous of degree one and $X_0$ is homogeneous of degree two. In this paper, we study the following equation with drift: $$ Lu\equiv\sum_{i,j=1}^{p_0}X_i(a_{ij}(x) X_ju) +a_0X_0u=\sum_{j=1}^{p_0}X_jF_j(x)\ , $$ where $a_{ij}(x)$ are real valued, bounded measurable functions defined in a domain $\Omega\subset G$, $a_{ij}(x)=a_{ji}(x)$, satisfying the uniform ellipticity condition in ${\mathbb R}^{p_0}$ and $a_0\in \mathbb{R}\backslash\{0\}$. Moreover, the coefficients $a_{ij}$ belong to the class $VMO$ (Vanishing Mean Oscillation) with respect to the subelliptic metric induced by the vector fields $X_0$, $X_1$, $X_2,\dots,X_{p_0}$. We derive local $L^p$ estimates for second order derivatives and H\"{o}lder estimates by establishing the representation formulas and higher order integrability of weak solutions to the above equation. Keywords: Homogeneous group, interior regularity, vector fields. MSC: 22E60, 35R03, 49N60 [ Fulltextpdf (387 KB)] for subscribers only. 