We deal with existence and regularity for weak solutions to Dirichlet problems of the type $$\begin{aligned} \left\{ \begin{array}{ll} - \mathrm{div} (A(x)Xu) +b(x)Xu + c(x)u=f\quad \hbox {in} \; \varOmega \\ \\ u=0 \quad \quad \hbox {on} \; \partial \varOmega . \end{array} \right. \end{aligned}$$in a bounded domain $$\varOmega $$ of $${\mathbb {R}}^n, n\ge 2.$$ We assume that the matrix of the coefficients $$A(x)= {^tA(x)}$$ satisfies the anisotropic bounds $$\begin{aligned} \frac{|\xi |^2}{K(x)}\le \langle A(x) \xi , \xi \rangle \le K(x) |\xi |^2\quad \quad \forall \xi \in {\mathbb {R}}^n,\; \hbox {for a.e.} \; x\in \varOmega \end{aligned}$$with the ellipticity function $$K(x)\in A_2\cap RH_{\tau }$$, $$\tau $$ opportunely related to the homogeneous dimension. The functions b(x) and c(x) are assumed to belong to some weighted Lebesgue spaces.

中文翻译：

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Carnot-Carathéodory空间中退化椭圆方程解的存在性和正则性

我们处理 $$\begin{aligned} \left\{ \begin{array}{ll} - \mathrm{div} (A(x)Xu) +b( x)Xu + c(x)u=f\quad \hbox {in} \; \varOmega \\ \\ u=0 \quad \quad \hbox {on} \; \partial \varOmega 。\end{array} \right。\end{aligned}$$ 在 $${\mathbb {R}}^n, n\ge 2.$$ 的有界域 $$\varOmega $$ 我们假设系数矩阵 $$A(x )= {^tA(x)}$$ 满足各向异性边界 $$\begin{aligned} \frac{|\xi |^2}{K(x)}\le \langle A(x) \xi , \ xi \rangle \le K(x) |\xi |^2\quad \quad \forall \xi \in {\mathbb {R}}^n,\; \hbox {for ae} \; x\in \varOmega \end{aligned}$$ 与椭圆函数 $$K(x)\in A_2\cap RH_{\tau }$$，$$\tau $$ 恰巧与齐次维度相关。假设函数 b(x) 和 c(x) 属于某些加权 Lebesgue 空间。