Existence and regularity of the solutions to degenerate elliptic equations in Carnot-Carathéodory spaces

Abstract

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.