I'm not exactly sure how to categorize this question so sorry if it's in the wrong place. I'm trying to understand one part of a lemma given in a paper on invariant sets for PDEs and I'm getting hung up on signs. I think it might just be me not understanding notation or forgetting some analysis. Anyway it looks like:
Let $G:\Bbb{R}^n \rightarrow \Bbb{R}$ be a smooth function, and let $dG$ and $d^2G$ denote the first and second derivatives of $G$. They note here that "the $v$ dependence of $dG$ and $d^2G$ is being suppressed." $v$ being a function of $x$ and $t$ that satisfies a generic form PDE: $$ \frac{\partial v}{\partial t}=\epsilon D \Delta v +\Sigma M^i \frac{\partial v}{\partial x^i}+f,\quad\epsilon>0. $$ And satisfies some boundary conditions. Continuing, with $\gamma\left(t\right)$ being a smooth curve in $\Bbb{R}^n$ and with dots denoting time derivatives, they note that: $$ \begin{split} \dot{\left(G\circ v \right)} &=dG\left(\dot{\gamma}\left(t\right) \right)\\ \ddot{\left(G\circ v \right)}&=dG^2\left(\dot{\gamma}\left(t\right) \right) +dG\left(\ddot{\gamma}\left(t\right) \right) \end{split}$$
Finally, suppose $v:\Bbb{R}^m \rightarrow \Bbb{R}^n$ is a smooth function with range in the set $\Sigma=\{v:G\left(v\right)\leq 0\}$ and that $G\circ v\left(x_0\right)=0$. Then for $\xi\in\Bbb{R}^m$ and $h\in\mathbb{R}$, $G\circ v\left(x_0+h\xi \right)\leq G\circ v\left(x_0\right)$. After this they make use of a Taylor expansion to show $dG\left(dv\left(\xi\right) \right)=0$ as $h$ goes to $0$.
I'm not sure if the information above the last paragraph is helpful, but I figured I'd include just in case. They use some of it later with a convexity argument to reach more conclusions. What's mixing me up here is how the range of $v$ is defined. I understand it to mean that $v$ maps points in $\Bbb R^m$ to points in $\Bbb R^n$ that, when $G$ is applied to them, give positive real numbers. So, if the value of $G\circ v\left(x_0\right)=0$ and I move in some direction $\xi$ and distance $h$ away from $x_0$, I would expect that that value is also going to be $\geq 0$ given the restrictions on the range of $v$. However, the inequality they give suggests that the quantity is $\leq 0$. Can you clarify?
Sorry for the long post.