Solving the $\mathrm{\partial }\bar{\mathrm{\partial }}$ for extendable currents without vanishing the boundary cohomology group

In this paper, we consider the problem of solving the $\mathrm{\partial }\bar{\mathrm{\partial }}$ equation with discribed support for differential forms in a relatively compact domain $\mathrm{\Omega }$ of a complex analytic manifold $X$. And as a consequence, we have the solution of the equation $\mathrm{\partial }\bar{\mathrm{\partial }}$ for extendable currents without the annulation assumption of the De Rham cohomology group of the boundary.\ $\mathbf{\text{Keywords:}}$ operator $\mathrm{\partial }\bar{\mathrm{\partial }}$, De Rham cohomology group, Dolbeault cohomology group, Bott-Chern cohomology group,Applie cohomology group, discribed support, extendable currents.