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The paper presents a first step towards a family index theorem for classicalself-adjoint boundary value problems. We address here the simplest non-trivialcase 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 orderself-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. Thesecond result is the universality of the index: we show that the index is auniversal additive homotopy invariant for such families, if the vanishing onfamilies of invertible operators is required.

Self-adjoint local boundary problems on compact surfaces. II. Family index