Abstract : In the first part, we consider a domain $\Omega$ with Lipschitz boundary, is relatively compact in an $n$-dimensional Kähler manifold, which satisfies some ``log \delta-pseudoconvexity'' conditions. We show that the $\overline\partial$ -equation with exact support in $\Omega$ admits a solution in bidegrees $(p,q)$, $1\leq q\leq n-1$. Moreover, the range of $\overline\partial$ acting on smooth $(p,n-1)$-forms with support in $\overline\partial$ is closed. Applications are given to the solvability of the tangential Cauchy-Riemann equations for smooth forms and currents for all intermediate bidegrees on boundaries of weakly pseudoconvex domains in Stein manifolds and to the solvability of the tangential Cauchy-Riemann equations for currents on Levi-flat $CR$ manifolds of arbitrary codimension. In a second part, we study the $\overline\partial$ -equation with zero Cauchy data along a hypersurface with constant signature. Applications to the solvability of the tangential Cauchy-Riemann equations for smooth forms with compact support and currents on the hypersurface are given. In particular the Hartogs phenomenon holds in weakly 2-convex-concave hypersurfaces with constant signature in Stein manifolds.