A degenerate Sobolev inequality for a large open set in a homogeneous space

Abstract:

In current literature, existence results for degenerate elliptic equations with rough coefficients on a large open set $\Theta$ of a homogeneous space $(\Omega ,d)$ have been demonstrated; see the paper by Gutierrez and Lanconelli (2003). These results require the assumption of a Sobolev inequality on $\Theta$ of the form \begin{equation*} (1)\qquad \qquad \Big \{\displaystyle {\int _\Theta } |w(x)|^{2\sigma }dx\Big \}^\frac {1}{2\sigma } \leq C \Big \{ \displaystyle {\int _\Theta } \mathcal {Q}(x,\nabla w(x))dx\Big \}^\frac {1}{2}, \qquad \qquad \qquad \qquad \end{equation*} holding for $w\in Lip_0(\Theta )$ and some $\sigma \in (1,2]$. However, it is unclear when such an inequality is valid, as techniques often yield only a local version of (1):

(2) \begin{equation*} \Big \{\displaystyle {\frac {1}{|B_r|}} \displaystyle {\int _{B_r}}|v(x)|^{2\sigma }\Big \}^\frac {1}{2\sigma } \leq Cr\Big \{\displaystyle {\frac {1}{|B_r|}}\displaystyle {\int _{B_r}} \mathcal {Q}(x,\nabla v(x))dx+\displaystyle {\frac {1}{|B_r|}} \displaystyle {\int _{B_r}}|v(x)|^2dx\Big \}^\frac {1}{2}, \end{equation*} holding for $v\in Lip_0(B_r)$, with $\sigma$ as above. The main result of this work shows that the global Sobolev inequality (1) can be obtained from the local Sobolev inequality (2) provided standard regularity hypotheses are assumed with minimal restrictions on the quadratic form $\mathcal {Q}(x,\cdot )$. This is achieved via a new technique involving existence of weak solutions, with global estimates, to a 1-parameter family of Dirichlet problems on $\Theta$ and a maximum principle.