Suppose $\{u_n\}$ be a sequence of functions in $C_c^\infty(\Omega)$, $\Omega$ is a bounded open set in $\mathbb R^N$. Is the gradient $\nabla u_n$ is uniformly bounded? namely, there exists constant $M$ independent of $n$ such that $||\nabla u_n||_\infty\leq M?$
2026-03-26 02:41:33.1774492893
uniform bound of gradient
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Certainly not. For instance, fix any $f\in C_c^\infty(\Omega)$ such that $\nabla f\neq 0$ and let $u_n=nf$ for each $n$.