Cohomology vanishing theorems for free boundary submanifolds

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.