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As the proof of **Theorem 2** suggests, this follows immediately from **Theorem 2.1**. While it's a bit ambiguously worded, to apply **Theorem 2.1**, we need $b_1(x)=0$ in $\Delta u + b_1(x)u_{x_1} + f(u) = 0$, $u > 0$ in $\Omega$, $u = 0$ on a part of $\partial\Omega$, and some basic continuity conditions. The assumptions of **Theorem 2** directly satisfies all of these.

Switching/modernizing notation a bit, I'll write $\nabla$ for the vector derivative which is the gradient when applied to a scalar function, i.e. $\nabla\phi = \operatorname{grad}\phi$. The directional derivative in the direction $\mathbf v$, where $\mathbf v$ is a vector, can be written as $\mathbf v \cdot \nabla$. For a coordinate, $x_i$, the corresponding vector $\mathbf e_i = \nabla x^i$ where $x^i(x_1,\dots, x_n) = x_i$. In other words, $\frac{\partial}{\partial x_i} = \mathbf e_i\cdot\nabla$. For completeness, though it won't be relevant, $\Delta = \nabla\cdot\nabla = \nabla^2$.

In the proof of **Theorem 2**, we are aligning the $x_1$ axis with the vector $\gamma$. That is, $\frac{\partial}{\partial x_1} = \gamma \cdot \nabla$, the directional derivative in the $\gamma$ direction. Therefore, $\gamma \cdot \operatorname{grad} u = \gamma \cdot \nabla u = \frac{\partial u}{\partial x_1} = u_{x_1}$. One of the first consequences of **Theorem 2.1** is that $u_{x_1} < 0$ in $\Sigma=\Sigma_\gamma$. **Theorem 2.1** also has $\Omega = \Sigma \cup \Sigma' \cup (T_{\lambda_1}\cap \Omega)$ if $\gamma \cdot \nabla u = 0$ at some point of $\Omega \cap T_{\lambda_1}$ as a conclusion. For the annulus, the $T_{\lambda_1}$ for varying $\gamma$ will be the lines tangent to the circle halfway into the annulus. Clearly the (closure of the) maximal cap unioned with its reflection is not all of the annulus which is what **Theorem 2.1** implies in this case, thus it can't be the case that $\gamma\cdot\nabla u = 0$ even when $|x| = (R'+R)/2$ where before we only knew this for $|x| > (R'+R)/2$.

The proof of **Theorem 2** is assuming that **Theorem 2.1** holds. The proof of **Theorem 2.1** doesn't come until later in the paper, and it's moderately involved.