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Coercivity and Gamma-convergence of the $p$-energy of sphere-valued Sobolev maps

Michele Caselli, Mattia Freguglia, Nicola Picenni

TL;DR

This work analyzes sphere-valued Sobolev maps $u_p$ with a uniform bound on the rescaled $p$-energy $(n-p)\int |\nabla u_p|^{p}$ as $p\uparrow n$, establishing flat convergence of the distributional Jacobians to an integral $m$-boundary and proving a Gamma-convergence result for the energy to the mass of the limit current. The authors develop a fully variational approach, including a zero-dimensional core estimate and a deformation-to-grid strategy to extend compactness and lower-bound controls to general dimension and codimension, yielding an integral, area-minimizing limit in a cobordism class under boundary data. They also provide a recovery sequence construction for the upper bound, connect their results to Hardt–Lin-type convergence for energy-minimizing maps, and obtain new energy estimates for prescribed singularities, all while highlighting parallels and distinctions with the Ginzburg–Landau and Yang–Mills–Higgs Gamma-convergence frameworks. The results advance understanding of singular limits in non-diffuse variational problems and offer tools for analyzing convergence of critical points and min-max sequences in high codimension. The framework accommodates Dirichlet boundary conditions and fractional-approach perspectives, underscoring the integrality and cobordism-prerequisites of the limiting currents.

Abstract

We consider sequences of maps from an $(n+m)$-dimensional domain into the $(n-1)$-sphere, which satisfy a natural $p$-energy growth, as $p$ approaches $n$ from below. We prove that, up to subsequences, the Jacobians of such maps converge in the flat topology to an integral $m$-current, and that the $p$-energy Gamma-converges to the mass of the limit current. As a corollary, we deduce that the Jacobians of $p$-energy minimizing maps converge to an integral $m$-current that is area-minimizing in a suitable cobordism class, depending on the boundary datum. Moreover, we obtain new estimates for the minimal $p$-energy of maps with prescribed singularities.

Coercivity and Gamma-convergence of the $p$-energy of sphere-valued Sobolev maps

TL;DR

This work analyzes sphere-valued Sobolev maps with a uniform bound on the rescaled -energy as , establishing flat convergence of the distributional Jacobians to an integral -boundary and proving a Gamma-convergence result for the energy to the mass of the limit current. The authors develop a fully variational approach, including a zero-dimensional core estimate and a deformation-to-grid strategy to extend compactness and lower-bound controls to general dimension and codimension, yielding an integral, area-minimizing limit in a cobordism class under boundary data. They also provide a recovery sequence construction for the upper bound, connect their results to Hardt–Lin-type convergence for energy-minimizing maps, and obtain new energy estimates for prescribed singularities, all while highlighting parallels and distinctions with the Ginzburg–Landau and Yang–Mills–Higgs Gamma-convergence frameworks. The results advance understanding of singular limits in non-diffuse variational problems and offer tools for analyzing convergence of critical points and min-max sequences in high codimension. The framework accommodates Dirichlet boundary conditions and fractional-approach perspectives, underscoring the integrality and cobordism-prerequisites of the limiting currents.

Abstract

We consider sequences of maps from an -dimensional domain into the -sphere, which satisfy a natural -energy growth, as approaches from below. We prove that, up to subsequences, the Jacobians of such maps converge in the flat topology to an integral -current, and that the -energy Gamma-converges to the mass of the limit current. As a corollary, we deduce that the Jacobians of -energy minimizing maps converge to an integral -current that is area-minimizing in a suitable cobordism class, depending on the boundary datum. Moreover, we obtain new estimates for the minimal -energy of maps with prescribed singularities.

Paper Structure

This paper contains 18 sections, 20 theorems, 216 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Motivation and related works
  4. The Ginzburg-Landau and Yang-Mills-Higgs frameworks
  5. The p-energy framework
  6. Authors' note
  7. The fractional framework
  8. Organization of the paper
  9. Definitions and notation
  10. Currents and flat norm
  11. Jacobians
  12. Quantitative approximation of Sobolev maps and their Jacobians
  13. The zero-dimensional case m=0
  14. Proof of the main result in general dimension and codimension
  15. Compactness and lower bound
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $n\geq 2$ and $m\geq 0$ be integers, and let $\Omega\subset \mathbb R^{n+m}$ be a bounded open set with Lipschitz boundary. For a (discrete) sequence $p\in (n-1,n)$ let $u_p \in W^{1,p}(\Omega;\mathbb{S}^{n-1})$ be maps such that Then the following statements hold.

Theorems & Definitions (40)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6: Theorem 3.1 in Lin-2011
  • Remark 2.1
  • Proposition 2.2: Theorem 3.8 in ABO2-singularities and Theorem 4.9 and Theorem 4.10 in Bre-Mi
  • Theorem 2.3: see Theorem 1 in BN-Invent and Corollary 1.7 in Bre-Mi
  • Lemma 3.1
  • ...and 30 more