Asymptotic behavior of minimizing $p$-harmonic maps when $p\nearrow 2$ in dimension 2

Abstract

We study $p$--harmonic maps with Dirichlet boundary conditions from a planar domain into a general compact Riemannian manifold. We show that as $p$ approaches $2$ from below, they converge up to a subsequence to a minimizing singular renormalizable harmonic map. The singularities are imposed by topological obstructions to the existence of harmonic mappings; the location of the singularities being governed by a renormalized energy. Our analysis is based on lower bounds on growing balls and also yields some uniform weak-$L^p$ bounds (also known as Marcinkiewicz or Lorentz $L^{p,\infty }$).

40 pages, typographical corrections

References in corpus (2)

• Topological singular set of vector-valued maps, II: $\Gamma$-convergence for Ginzburg-Landau type functionals
• Trace theory for Sobolev mappings into a manifold

Cited by in corpus (3)

• Limiting behavior of minimizing $p$-harmonic maps in 3d as $p$ goes to $2$ with finite fundamental group
• Stationary $p$-harmonic maps approaching planar singular harmonic maps to the circle
• Universality of renormalisable mappings in two dimensions: the case of polar convex integrands