Γ-convergence of the p-Dirichlet energy for manifold-valued maps

2025/05/27 by Giacomo Canevari, Van Phu Cuong Le, Canevari, Giacomo +5
Computer Science · Mathematics · #49Q15 #49Q20 #58E12 #58E20 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2505.21257

openalex publication_date 2025/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29

Abstract

We prove a Γ-convergence result for the p-Dirichlet energy functional defined on maps from a smooth bounded domain Ω⊆ ℝ^n+k to \mathscrN, a (k-2)-connected and smooth closed Riemannian manifold with Abelian fundamental group, where n and k are integers, n ≥ 0, k ≥ 2. We focus on the regime p →~k^- under Dirichlet boundary conditions. The result provides a description of the asymptotic behavior of the topological singular sets for families of \mathscrN-valued Sobolev maps which satisfy suitable energy bounds. Such topological singular sets are n-dimensional flat chains with coefficients in π_k-1(\mathscrN) endowed with a suitable norm. As a consequence of our main result, it follows that the topological singular sets of energy minimizing p-harmonic maps converge to a n-dimensional flat chain S with coefficients in π_k-1(\mathscrN) which has finite mass and solves the Plateau problem within the homology class associated to the boundary datum.

Citations

• Coercivity and Gamma-convergence of the p-energy of sphere-valued Sobolev maps
• A nonlocal approximation of the area in codimension two
• Energy-minimizing torus-valued maps with prescribed singularities, Plateau's problem, and BV-lifting
• Energy minimizing maps with prescribed singularities and Gilbert-Steiner optimal networks
• Mountain pass energies between homotopy classes of maps
• p-Harmonic Maps to S¹ and Stationary Varifolds of Codimension 2
• Topological singular set of vector-valued maps, I: Applications to\n manifold-constrained Sobolev and BV spaces
• Discontinuous order parameters in liquid crystal theories
• Normal and Integral Currents

Related

• Coercivity and Gamma-convergence of the p-energy of sphere-valued Sobolev maps
• Critical domains for certain Dirichlet integrals in weighted manifolds
• Energy identity for Ginzburg-Landau approximation of harmonic maps
• Discrete and Continuous Green Energy on Compact Manifolds
• Energy estimates of harmonic maps between Riemannian manifolds