2021/10/17 by Giovanni Di Fratta, Di Fratta, Giovanni, Alberto Fıorenza +3
Mathematics · #35A23 #35R45 #49R05 #49S05 #82D40 #Advanced Differential Equations and Dynamical Systems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Pattern Formation and Solitons (nlin.PS)
paper · pdf · doi:10.48550/arxiv.2110.08755
openalex publication_date 2021/10/17 · openalex created_date 2022/10/12 · openalex updated_date 2026/07/28
The paper concerns the analysis of global minimizers of a Dirichlet-type energy functional in the class of \mathbbS2-valued maps defined in cylindrical surfaces. The model naturally arises as a curved thin-film limit in the theories of nematic liquid crystals and micromagnetics. We show that minimal configurations are z-invariant and that energy minimizers in the class of weakly axially symmetric competitors are, in fact, axially symmetric. Our main result is a family of sharp Poincaré-type inequality on the circular cylinder, which allows establishing a nearly complete picture of the energy landscape. The presence of symmetry-breaking phenomena is highlighted and discussed. Finally, we provide a complete characterization of in-plane minimizers, which typically appear in numerical simulations for reasons we explain.