Let D be a Stein domain in the product manifold of an m-dimensional complex manifold Z and an n-dimensional Stein manifold S. Let \({\mathcal O}\) be the sheaf over D of germs of holomorphic functions and \({\mathcal O}^ p\!_{z,s}\) be the sheaf of p-forms \(g=\sum_{1\leq i_ 1<i_ 2<...<i_ p\leq m}g_{i_ 1i_ 2...i_ p}(z,s)dz_{i_ 1}\wedge dz_{i_ 2}\wedge...\wedge dz_{i_ p},\) where \(g_{i_ 1i_ 2...i_ p}(z,s)'s\) are elements of \({\mathcal O}\), and \(d^ p\!_ z:{\mathcal O}^ p\!_{z,s}\to {\mathcal O}^{p+1}\!\!\!_{z,s}\) is the complex differential with respect to the variable z. Assume that D has a regular boundary and finitely generated \({\mathbb{C}}\)-cohomology. Then the following properties (1) and (2) are equivalent: (1) For any \(p=1,2,...,m\) and \(g\in H^ 0(D,{\mathcal O}^ p\!_{z,s})\) satisfying \(d^ p\!_ zg=0\), there exists a global form \(f\in H^ 0(D,{\mathcal O}^{p- 1}\!\!\!_{z,s})\) satisfying \(d_ z\!^{p-1}f=g\). - (2) For any (z,s)\(\in D\), we have \(H^ p(D(z,s),{\mathbb{C}})=0\) for \(p=1,2,...,m\), the space \(\tilde D\) of all connected components \(\psi^{-1}(s)\), (z,s) running over D and \(\psi:D\to S\) being the canonical mapping, is a Hausdorff space and, for the natural complex structure, \(\tilde D\) is a Stein manifold. - Since the above property (2) is local in the parameter space S, we have a local characterization of the global existence (1).