Let $\Omega\subset\mathbb{R}^n $ be open, bounded and connected and suppose that $\Omega$ satisfies the interior ball regularity at every point in $\partial \Omega$. For $x_0 \in \partial \Omega$ denote $\nu(x_0)$ the exterior normal to an interior balltangent to $\partial \Omega$ at $x_0$. Consider $$Lu:=-\sum_{i,j=1}^{n} a_{ij}u_{x_i x_j} + \sum_{i=1}^{n} b_i u_{x_i}+ cu$$ where $a_{ij}, b_{i}, c \in C(\overline{\Omega})$, $c\ge 0$ and $a_{ij}$ are unformly elliptic. Suppose $u \in C^{2} (\Omega) \cap C(\overline{\Omega})$ satisfies $Lu=0$.
(1) Assume that $\partial \Omega=\partial_{D}\Omega\cup \partial_{N}\Omega$ , $\partial_{D}\Omega \neq \emptyset$ and $u \in C^{1}(\Omega \cup \partial_{N}\Omega)$ satisfies $$u=0 \ \text{on} \ \partial_{D}\Omega \ \text{and} \sum_{i=1}^{n}\beta_{i}(x)u_{x_i}=0 \ \text{on} \ \partial_{N}\Omega$$ where $\beta(x)=(\beta_1(x) , \ldots , \beta_n(x))$ has a nonzero normal component (to the interior ball) at each $ x\in \partial_{N}\Omega$. Prove that $u \equiv 0$.
(2) If $u \in C^1(\overline{\Omega})$ satisfies tehe oblique boundary condition $$\alpha(x)u+\sum_{i=1}^n \beta_i(x)u_{x_i}=0 \ \text{on} \ \partial \Omega$$ where $(\beta \cdot \nu)\alpha>0$, then prove that $u \equiv 0$.