Michell trusses in two dimensions as a Gamma-limit of optimal design problems in linear elasticity

We reconsider the minimization of the compliance of a two dimensional elastic body with traction boundary conditions for a given weight. It is well known how to rewrite this optimal design problem as a nonlinear variational problem. We take the limit of vanishing weight by sending a suitable Lagrange multiplier to infinity in the variational formulation. We show that the limit, in the sense of $\Gamma$-convergence, is a certain Michell truss problem. This proves a conjecture by Kohn and Allaire.