  Let $L$ be a special Lagrangian submanifold of a compact, Calabi-Yau manifold
$M$ with boundary lying on the symplectic, codimension 2 submanifold $W$. It is
shown how deformations of $L$ which keep the boundary of $L$ confined to $W$
can be described by an elliptic boundary value problem, and two results about
minimal Lagrangian submanifolds with boundary are derived using this fact. The
first is that the space of minimal Lagrangian submanifolds near $L$ with
boundary on $W$ is found to be finite dimensional and is parametrised over the
space of harmonic 1-forms of $L$ satisfying Neumann boundary conditions. The
second is that if $W'$ is a symplectic, codimension 2 submanifold sufficiently
near $W$, then under suitable conditions, there exists a minimal Lagrangian
submanifold $L'$ near $L$ with boundary on $W'$.
