Shape optimization of light structures and the vanishing mass conjecture

This work proves rigorous results about the vanishing mass limit of classical problem to find a shape with minimal elastic compliance. Contrary all previous in mathematical literature, which utilize soft constraint by introducing Lagrange multiplier, we here consider hard constraint. Our are first establish convergence approximately optimal shapes (exact) size ??0 generalized represented (possibly diffuse) probability measure. is minimizer compliance, involves new integrand, namely one conjectured Bouchitté 2001 and predicted heuristically before works Allaire, Kohn, Strang from 1980s 1990s. integrand gives energy understood as fine oscillation (optimal) lower-dimensional structures. Its appearance surprising since original compliance just quadratic form nonconvexity not immediately obvious. In fact, it interaction requirement attaining loading (in divergence constraint) that rise this integrand. We also present connections theory Michell trusses, formulated 1904, show how our can be interpreted justification on level functionals both two three dimensions, settling open problem. proofs rest compensated compactness arguments applied an explicit family (symmetric) div-quasiconvex forms, computations involving Hashin–Shtrikman bounds for Kohn–Strang characterization minimizers due Buttazzo.