Limiting Behavior of Minimizing p-Harmonic Maps in 3d as p Goes to 2 with Finite Fundamental Group

Abstract We study the limiting behavior of minimizing p -harmonic maps from a bounded Lipschitz domain $$\Omega \subset \mathbb {R}^{3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> </mml:math> to a compact connected Riemannian manifold without boundary and with finite fundamental group as $$p \nearrow 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>↗</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . We prove that there exists a closed set $$S_{*}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mi>S</mml:mi> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> <mml:mrow/> </mml:mmultiscripts> </mml:math> of finite length such that minimizing p -harmonic maps converge to a locally minimizing harmonic map in $$\Omega \setminus S_{*}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>\</mml:mo> <mml:mmultiscripts> <mml:mi>S</mml:mi> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> <mml:mrow/> </mml:mmultiscripts> </mml:mrow> </mml:math> . We prove that locally inside $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> the singular set $$S_{*}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mi>S</mml:mi> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> <mml:mrow/> </mml:mmultiscripts> </mml:math> is a finite union of straight line segments, and it minimizes the mass in the appropriate class of admissible chains. Furthermore, we establish local and global estimates for the limiting singular harmonic map. Under additional assumptions, we prove that globally in $$\overline{\Omega }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>Ω</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> the set $$S_{*}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mi>S</mml:mi> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> <mml:mrow/> </mml:mmultiscripts> </mml:math> is a finite union of straight line segments, and it minimizes the mass in the appropriate class of admissible chains, which is defined by a given boundary datum and $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> .