Calculus of Variations and Geometric Measure Theory

S. Dovetta - L. Tentarelli

Symmetry breaking in two-dimensional square grids: persistence and failure of the dimensional crossover

created by tentarelli on 07 Jul 2022


Published Paper

Inserted: 7 jul 2022
Last Updated: 7 jul 2022

Journal: J. Math. Pures Appl.
Volume: 160
Pages: 99-157
Year: 2022
Doi: 10.1016/j.matpur.2021.12.010

ArXiv: 2104.13842 PDF


We discuss the model robustness of the infinite two-dimensional square grid with respect to symmetry breakings due to the presence of defects, that is, lacks of finitely or infinitely many edges. Precisely, we study how these topological perturbations of the square grid affect the so-called dimensional crossover identified in Adami et al. 2019. Such a phenomenon has two evidences: the coexistence of the one and the two-dimensional Sobolev inequalities and the appearence of a continuum of $L^2$-critical exponents for the ground states at fixed mass of the nonlinear Schr\"odinger equation. From this twofold perspective, we investigate which classes of defects do preserve the dimensional crossover and which classes do not.