This paper is devoted to studying some applications of the Bochner-Kodaira-Morrey-Kohn identity. For this study, we define a condition which is called $(H_q)$ condition which is related to the Levi form on the complex manifold. Under the $(H_q)$ condition and combining with the basic Bochner-Kodaira-Morrey-Kohn identity, we study the $L^2$ $\overline\partial$ Cauchy problems on domains in $\mathbb{C}^{n}$, Kähler manifold and in projective space. Also, we study this problem on a piecewise smooth strongly pseudoconvex domain in a complex manifold. Furthermore, the weighted $L^{2}$ $\overline\partial$ Cauchy problem is studied under the same condition in a Kähler manifold with semi-positive holomorphic bisectional curvature. On the other hand, we study the global regularity and the $L^2$ theory for the $\overline\partial$-operator with mixed boundary conditions on an annulus domain in a Stein manifold between an inner domain which satisfy $(H_{n-q-1})$ and an outer domain which satisfy $(H_q)$.