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Liftings of Sobolev maps into closed Riemannian manifolds via double coverings and minimal connections relative to planar sets, with an application to ferronematics

Giacomo Canevari, Federico Luigi Dipasquale, Bianca Stroffolini

Abstract

We consider Sobolev maps from a planar domain into a closed Riemannian manifold and their BV liftings via a double covering of the target. We establish a sharp lower bound on the jump length of the lifting, expressed in terms of a geometric quantity: the minimal connection, relative to the domain, of the non-orientable singularities. As an application, we analyse minimisers of a two-dimensional model of ferronematics under ``mixed'' boundary conditions -- that is, Dirichlet conditions for the liquid crystal order parameter and Neumann conditions for the magnetisation vector.

Liftings of Sobolev maps into closed Riemannian manifolds via double coverings and minimal connections relative to planar sets, with an application to ferronematics

Abstract

We consider Sobolev maps from a planar domain into a closed Riemannian manifold and their BV liftings via a double covering of the target. We establish a sharp lower bound on the jump length of the lifting, expressed in terms of a geometric quantity: the minimal connection, relative to the domain, of the non-orientable singularities. As an application, we analyse minimisers of a two-dimensional model of ferronematics under ``mixed'' boundary conditions -- that is, Dirichlet conditions for the liquid crystal order parameter and Neumann conditions for the magnetisation vector.
Paper Structure (15 sections, 19 theorems, 99 equations, 6 figures)

This paper contains 15 sections, 19 theorems, 99 equations, 6 figures.

Table of Contents

  1. Liftings and relative minimal connections
  2. Double coverings of Riemannian manifolds
  3. Euclidean embeddings and function spaces.
  4. Orientable and non-orientable singularities.
  5. Sobolev liftings.
  6. Properties of minimal relative connections
  7. Liftings and sets of finite perimeter
  8. Proof of Proposition \ref{['prop:minconn-finiteper']}
  9. Relative minimal connections in a model of ferronematics
  10. Preliminary estimates and notation
  11. The Euler-Lagrange equations and the maximum principle.
  12. Notation on the Ginzburg-Landau functional.
  13. Allen-Cahn structure for the $\mathbf{M}$-component.
  14. Proof of Theorem \ref{['th:ferronematics']}
  15. Acknowledgments

Key Result

Theorem 1

Let $\mathscr{N}$, $\mathscr{E}$ be closed, smooth Riemannian manifolds and $\Pi\colon\mathscr{E}\to\mathscr{N}$ a smooth double covering map. Let $\Omega\subseteq\mathbb{R}^2$ be a bounded, simply connected domain of class $C^2$ and let $a_1$, …, $a_p$, $b_1$, …, $b_r$ be distinct points in $\Omega be a map with a non-orientable singularity at each $a_j$ and an orientable singularity at each $b_h

Figures (6)

  • Figure 1: Example of a connection for $\{a_1 \, \ldots, a_6\}$ relative to $\Omega$.
  • Figure 2: Illustration of the proof of Lemma \ref{['lemma:mindisj']}.
  • Figure 3: The set $\Omega_\rho$ introduced in the proof of Lemma \ref{['lemma:goodlifting']} (in blue). The set $G_\rho$ is the union of $\Omega_\rho$ and the red regions.
  • Figure 4: The set $\mathscr{L}_A$ defined in \ref{['L_A']}. In this example, $A$ is the set in red and $\mathscr{L}_A$ consists of two polygonal paths (in blue) joining $a_1$ with $a_3$ and $a_2$ with $a_4$, as well as a finite union of loops.
  • Figure 5: A set $A$ (in red) and its arcs. The essential arcs are indicated by solid lines, the non-essential ones by dashed lines.
  • ...and 1 more figures

Theorems & Definitions (36)

  • Theorem 1
  • Theorem 2
  • Remark 1.1
  • Lemma 1.1
  • proof : Proof of Lemma \ref{['lemma:Xi']}
  • Proposition 1.2: SchoenUhlenbeck2
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • proof
  • ...and 26 more