Uniform and Lipschitz continuity of objective functions in metamaterial band gap optimization problems

In the last years, various articles have dealt with the analysis of the Floquet-Bloch spectrum of periodic metamaterials containing resonators, and the optimization of selected acoustic band gaps between consecutive dispersion surfaces in that spectrum. Applications include opening/enlarging/closing/shifting band gaps in target acoustic frequency ranges. This work investigates uniform and Lipschitz continuity of the objective functions of the resulting optimization problems (focusing on the case of band gap maximization), and their consequences for solving them numerically with performance guarantees. This sheds light on numerical results obtained by the authors in previous works.