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

## TL;DR

This paper proves that as the weight constraint in a 2D linear elasticity optimal design problem vanishes, the solutions converge to a Michell truss structure, confirming a longstanding conjecture.

## Contribution

It establishes the Gamma-limit of the optimal design problem as a Michell truss, providing a rigorous mathematical link between elasticity optimization and truss structures.

## Key findings

- Gamma-convergence of the optimal design problem to Michell trusses
- Confirmation of Kohn and Allaire's conjecture
- Mathematical characterization of the limit structure

## Abstract

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.