Let $L$ be a uniformly elliptic operator with $c \equiv 0$ on a bounded domain $U \subset \mathbb{R}^n$ with $C^2$-boundary, and let $\partial U = S_1 \cup S_2$. Suppose $u \in C^2 (\bar{U})$ satisfies the equation $Lu=0$ with boundary conditions $u=0$ on $S_1$ and $\frac{\partial u}{\partial n} = 0$ on $S_2$. Prove that $u \equiv 0$.
I actually have not idea on how starting with that problem. I thought about Hopf Boundary Point Lemma, but I'm stuck, I simply cannot go on. Any hint would be more than appreciated.
Thanks in advance!
Here we should add the condition $S1 \bigcap S2 = \emptyset$.
Since $Lu \ge 0$, by Hopf principle:$u$ can only attains itˊs non-positive minimum of $\overline{\Omega}$ on $x_0 \in \partial \Omega$, and $\frac{\partial u}{\partial n}(x_0) <0 $. Hence $x_0 \in S1$.
Similarly consider the $Lu \le 0$ condition, namely we have $u$ attains itˊs non-negative maximum of $\overline{\Omega}$ on $\partial \Omega$, say at point $x=x_1$. Hence $\frac{\partial u}{\partial n}(x_1) \gt 0$ and $x_1 \in S1$.
Consequently $u(x_0)=u(x_1)=0$ and $u \equiv 0$.