Conditions for the $\overline\partial$-closedness of differential forms

The $\overline\partial$-closed differential forms with smooth coefficients are studied in the closure of a bounded domain $D\subset\mathbb C^n$. It is demonstrated that the condition of $\overline\partial$-closedness can be replaced with a weaker differential condition in the domain and differential conditions on the boundary. In particular, for the forms with harmonic coefficients the $\overline\partial$-closedness is equivalent to some boundary relations. This allows us to treat the results as conditions for the $\overline\partial$-closedness of an extension of a form from the boundary.