On the existence of a function satisfying a certain inequality

Question

Let $D$ be an open disk in $\mathbb{R}^n$ ($n\ge 1$). I am wondering if there is a nonnegative function $f:D\rightarrow \mathbb{R}$ with $$f\in H^1(D)\cap L^\infty(D)$$ satisfying $$|\nabla f(x)|\le cf(x)$$ for $x\in G$, where $G$ is an open layer around the boundary of $D$, and $$f(x)=0$$ if $x$ is outside of $G$. I am trying to find an example at least in dimensions 1 and 2.

Answer 1

No, such a function cannot exist. In one-dimension, the inequality $|f'(x)|\leq c f(x)$ implies $f(x)\leq f(a) e^{cx}$ (Grönwall's lemma). Thus, if $f$ is nonnegative and zero somwhere, it must be zero everywhere. The higher dimensional case can be reduced to the one-dimensional case by restricting $f$ to lines.