  The paper presents a first step towards a family index theorem for classical
self-adjoint boundary value problems. We address here the simplest non-trivial
case of manifolds with boundary, namely the case of two-dimensional manifolds.
The first result of the paper is an index theorem for families of first order
self-adjoint elliptic differential operators with local boundary conditions,
parametrized by points of a compact topological space $X$. We compute the
$K^1(X)$-valued index in terms of the topological data over the boundary. The
second result is the universality of the index: we show that the index is a
universal additive homotopy invariant for such families, if the vanishing on
families of invertible operators is required.
