Robert V. Kohn, kohn@cims.nyu.edu
Co-authors: Benedikt Wirth
Optimal design revisited: The leading order effect of perimeter regularization
We consider a 2D shape optimization problem involving a structure loaded in shear. When the goal is to
minimize compliance, optimality requires microstructure and a 2nd rank laminate is known to suffice. We ask
the question: what happens when the goal is to minimize compliance plus epsilon times perimeter? Our main
accomplishment is to identify the leading-order effect of the perimeter regularization. This entails proving
upper and lower bounds that match (with respect to scaling). The construction associated with the upper bound
resembles a second-rank laminate, but uses branching rather than approximate interfaces. The ansatz-free lower
bound uses the Hashin–Shtrikman variational principle, and therefore works mainly in Fourier space