  We construct a manifestly supersymmetric and kappa-symmetry invariant
worldvolume action describing the coupling of a dynamical membrane to an
$\mathcal N=1$, $D=4$ $SU(N)$ super-Yang-Mills multiplet. Worldvolume scalar
fields in this action are a Goldstone and a Goldstino associated with
spontaneous breaking, by the membrane, of half of $\mathcal N=1$, $D=4$
supersymmetry. When the Goldstone fields are set to zero, the model reduces to
an $\mathcal N=1$, $d=3$ $SU(N)$ Chern-Simons theory induced by the SYM
coupling. We show that, when the membrane couples to the Veneziano-Yankielowicz
(VY) effective theory of the $\mathcal N=1$ SYM, it sources VY bulk field
equations, separates two distinct SYM vacua and provides the missing
contribution to the tension of BPS saturated domain-wall configurations, for
which the membrane serves as a core. As a result, we obtain explicit BPS
domain-wall solutions in the Veneziano-Yankielowicz theory. We also briefly
discuss a supersymmetric system of an open membrane having a string attached to
its boundary and coupled to a massive extension of the Veneziano-Yankielowicz
model.
