Equi-integrable approximation of Sobolev mappings between manifolds

TL;DR

The paper tackles the problem of when smooth maps are densely approximable in Sobolev spaces of maps between compact manifolds, focusing on equi-integrable sequences. It introduces the equi-integrable closure ${\mathrm{H}}^{1,p}_{\mathrm{Ei}}$ and proves its equality with the strong closure ${\mathrm{H}}^{1,p}_{\mathrm{St}}$ for the critical range $p\in\{1,\dots,\dim\mathcal M-1\}$, with extensions to higher-order and fractional spaces. The authors develop a skeleton- and VM0-homotopy framework to preserve homotopy under equi-integrable limits, and they connect these analytical results to cohomological obstructions via Jacobians, providing both local and global characterizations. The work thus unifies strong density results with topological criteria, yielding robust density theorems applicable to variational problems, harmonic maps, and nonlinear elasticity where maps between manifolds arise.

Abstract

We show that limits of sequences of smooth maps between compact Riemannian manifolds with equi-integrable $W^{1, p}$-Sobolev energy can always be strongly approximated by smooth maps, giving a counterpart of Hang's density result in $W^{1, 1}$ for the Sobolev space $W^{1, p}$ with integer $p \ge 2$. Our result extends to higher-order Sobolev spaces and is straightforward in fractional Sobolev spaces. We also provide a proof based on the weak continuity of Jacobians in the cases where the cohomological criterion of Bethuel, Demengel, Colon and Hélein applies.

Theorems & Definitions (52)

• Theorem 1.1
• Proposition 2.1
• Proposition 2.2
• proof : Proof of \ref{['proposition_homotopy_VMO_Rp']}
• Proposition 2.3
• proof : Proof of \ref{['proposition_homotopy_equiintegrable_Rp']}
• Proposition 2.4
• proof
• Proposition 2.5
• proof : Proof of \ref{['proposition_W1p_good_grid_measure']}