Calculus of Variations and Geometric Measure Theory

G. Di Fratta - A. Fiorenza - V. Slastikov

On symmetry of energy minimizing harmonic-type maps on cylindrical surfaces

created by difratta on 21 Feb 2021
modified on 14 Jun 2023


Published Paper

Inserted: 21 feb 2021
Last Updated: 14 jun 2023

Journal: Mathematics in Engineering
Year: 2021


The paper concerns the analysis of global minimizers of a Dirichlet-type energy functional in the class of $\mathbb{S}^2$-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 \emph{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.