  Harmonic functions $u:{\mathbb R}^n \to {\mathbb R}^m$ are equivalent to
integral manifolds of an exterior differential system with independence
condition $(M,{\mathcal I},\omega)$. To this system one associates the space of
conservation laws ${\mathcal C}$. They provide necessary conditions for
$g:{\mathbb S}^{n-1} \to M$ to be the boundary of an integral submanifold. We
show that in a local sense these conditions are also sufficient to guarantee
the existence of an integral manifold with boundary $g({\mathbb S}^{n-1})$. The
proof uses standard linear elliptic theory to produce an integral manifold
$G:D^n \to M$ and the completeness of the space of conservation laws to show
that this candidate has $g({\mathbb S}^{n-1})$ as its boundary. As a corollary
we obtain a new elementary proof of the characterization of boundaries of
holomorphic disks in ${\mathbb C}^m$ in the local case.
