  We investigate the properties of certain elliptic systems leading, a~priori,
to solutions that belong to the space of Radon measures. We show that if the
problem is equipped with a so-called asymptotic Uhlenbeck structure, then the
solution can in fact be understood as a standard weak solution, with one
proviso: analogously as in the case of minimal surface equations, the
attainment of the boundary value is penalized by a measure supported on (a
subset of) the boundary, which, for the class of problems under consideration
here, is the part of the boundary where a Neumann boundary condition is
imposed.
