On the existence of integrable solutions to nonlinear elliptic systems   and variational problems with linear growth
  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. 
Lisa Beck, Miroslav Bul\'i\v{c}ek, Josef M\'alek, Endre S\"uli
math.AP