  In this paper, via a new Hardy type inequality, we establish some cohomology
vanishing theorems for free boundary compact submanifolds $M^n$ with $n\geq2$
immersed in the Euclidean unit ball $\mathbb{B}^{n+k}$ under one of the
pinching conditions $|\Phi|^2\leq C$, $|A|^2\leq \widetilde{C}$, or $|\Phi|\leq
R(p,|H|)$, where $A$ $(\Phi)$ is the (traceless) second fundamental form, $H$
is the mean curvature, $C,\widetilde{C}$ are positive constants and $R(p,|H|)$
is a positive function. In particular, we remove the condition on the flatness
of the normal bundle, solving the first question, and partially answer the
second question on optimal pinching constants proposed by Cavalcante, Mendes
and Vit\'{o}rio.
