Robust structures in compliance minimization

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It is a classic engineering problem to identify which geometry of an elastic material best supports a load while consuming a minimum material amount. Typically, the optimal geometry would involve an infinitely fine microstructure with infinitely many, infinitely fine holes. Mathematically, the problem is either relaxed or regularized to admit the existence of the optimal solutions (energy minimizers). In this project we consider a weak form of regularization, which allows for existence of minimizers which nevertheless exhibit very fine (but finite) structures. Our aim is to study stability of these complicated structures, both in the deterministic as well as stochastic sense. In particular, we aim to deal with the uncertainty in the material parameters as well as loading scenarios, to understand robustness of these structures.