Let $\Omega$ be a bounded subset in $\mathbb{R}^n$, $n \ge 3$, with smooth boundary $\partial \Omega$. Assume given $a^{ij} \in W^{2, p}(\Omega)$ with $a^{ij} = a^{ji}$, $f \in L^p(\Omega)$ and $g \in L^q(\partial \Omega)$ with $p > n/2$ and $q > n-1$. Assume further that there exist constants $\lambda$ and $\Lambda$ such that, for all $x \in \Omega$ and $v \in \mathbb{R}^n$, we have $$ \lambda |v|^2 \le a^{ij} v_i v_j \le \Lambda |v|^2. $$
We consider the elliptic PDE $$ \begin{aligned} - \frac{\partial}{\partial x^j} \left( a^{ij} \frac{\partial u}{\partial x^i} \right) + f u &= h,\\ a^{ij} \nu_j \frac{\partial u}{\partial x^i} + g u &= 0, \end{aligned} $$ where $\nu$ is the unit outgoing normal to $\partial \Omega$ and $h \in L^r(\Omega)$ is given, $r \in (1, \infty)$.
I assume that the quadratic form $$ u \mapsto \frac{1}{2} \int_{\Omega} \left[a^{ij} \frac{\partial u}{\partial x^i} \frac{\partial u}{\partial x^j} + f u^2 \right] dx + \frac{1}{2}\int_{\partial \Omega} g u^2 dx $$ (naturally associated to this problem) is coercive on $H^1(\Omega)$. So, in particular, the homogeneous system (i.e. $h \equiv 0$) only admits the trivial solution $u \equiv 0$.
As is "well known" (any reference for this fact is welcome), we can write $$ u(x) = \int_{\Omega} G(x, y) h(y) dy $$ where $G$ is the associated Green function.
How can we prove, in such a low regularity context, that $G$ is bounded from below by a positive constant?
(Regularity for $g$ can be improved to e.g. $g \in W^{1-1/p, p}(\partial \Omega)$ if needed).
Any reply will be appreciated!
