Strong comparison principle for solutions of quasilinear equations

Abstract:

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 \[ (P_i)\quad -\mathrm {div} (\mathbf {a}(x,\nabla u_i)) = f_i \quad \mathrm {in } \Omega , \quad i=1,2.\] 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.