2025/12/29 by Coz, Stefan Le, Shakarov, Boris
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2512.23286
In this work, we study the dimensional reduction of stationary states in the shrinking limit for a broad class of two-dimensional domains, called open books, to their counterparts on metric graphs. An open book is a two-dimensional structure formed by rectangular domains sharing common boundaries. We first develop a functional-analytic framework suited to variational problems on open books and establish the existence of solutions as constrained action minimizers. For graph-based open books (i.e., those isomorphic to the product of a graph with an interval) we prove the existence of a sharp transition in the dimensionality of ground states. Specifically, there exists a critical transverse width: below this threshold, all ground states coincide with the ground states on the underlying graph trivially extended in the transverse direction; above it, ground states become genuinely two-dimensional.