Strong comparison principle for solutions of quasilinear equations

Let $\Omega \subset \mathbb{R} ^N$, $N \geq 1$, be a bounded smooth connected open set and $\mathbf{a} : \Omega \times \mathbb{R} ^N \to \mathbb{R} ^N$ be a map satisfying the hypotheses (H1)-(H4) below. Let $f_1,f_2 \in \mathrm{L}_{loc}^{1} (\Omega)$ with $f_2 \geq f_1$, $f_1 \not\equiv f_2$ in $\Omega$ and $u_1, u_2 \in \mathcal{C}^{1,\theta} (\overline \Omega)$ with $\theta \in (0,1]$ be two weak solutions of

Suppose that $u_2 \geq u_1$ in $\Omega$. Then we show that $u_2 > u_1$ in $\Omega$ under the following assumptions: either $u_2>u_1$ on $\partial \Omega$, or $u_1=u_2=0$ on $\partial \Omega$ and $u_1 \geq 0$ in $\Omega$. We also show a measure-theoretic version of the Strong Comparison Principle.