Let $u$ be the first Dirichlet eigenfunction, does there exist $\epsilon,\delta>0$ such that $|\nabla u(x)|\geq\epsilon$ for any $x$ st $u(x)<\delta$

Question

Let $\Omega\subset \mathbb{R}^2$ be a strictly convex region with diameter equals to 2. Let $u$ be the first positive Dirichlet eigenfunction and assume further that $u\leq 1$. I want to prove that, there exists $\delta,\epsilon>0$ such that $|\nabla u(x)|\geq\epsilon$ for any $x\in\Omega\backslash\Omega_\delta$, where $\Omega_\delta=\{x\in\Omega:u(x)\geq\delta\}$.

Actually I have no idea on this proposition. Usually I'm used to derive some estimate in the integral sense, but how to derive a lower bound estimation pointwisely?

Any help will be appreciated a lot!