  We establish a general framework for the analysis of boundary value problems
of matrix models at zero energy on compact regions. We derive existence and
uniqueness of ground state wavefunctions for the mass operator of the $D=11$
regularized supermembrane theory, that is the $\mathcal{N}=16$ supersymmetric
$SU(N)$ matrix model, on balls of finite radius. Our results rely on the
structure of the associated Dirichlet form and a factorization in terms of the
supersymmetric charges. They also rely on the polynomial structure of the
potential and various other supersymmetric properties of the system.
