p -Harmonic maps to $$S^1$$ S 1 and stationary varifolds of codimension two
- PDF / 604,946 Bytes
- 47 Pages / 439.37 x 666.142 pts Page_size
- 33 Downloads / 155 Views
Calculus of Variations
p-Harmonic maps to S 1 and stationary varifolds of codimension two Daniel Stern1 Received: 15 May 2019 / Accepted: 31 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We study the limiting behavior as p ↑ 2 of the singular sets Sing(u p ) and p-energy measures μ p := (2 − p)|du p | p dvol for families of stationary p-harmonic maps u p ∈ W 1, p (M, S 1 ) from a closed, oriented manifold M to the circle. When the measures μ p have uniformly bounded mass, we show that—up to subsequences—the singular sets Sing(u p ) converge in the Hausdorff sense to the support of a stationary, rectifiable varifold V of codimension 2, and the measures μ p converge weakly in (C 0 (M))∗ to a limit of the form μ = V + |h|2 dvol, where h is a harmonic one-form. For solutions on two-dimensional domains, we show moreover that the density of V takes values in 2πN. Finally, we observe that nontrivial families u p of such maps arise naturally on any closed Riemannian manifold of dimension n ≥ 2, via variational methods. Mathematics Subject Classification 53C43 · 58E20
1 Introduction In their 1995 paper [19], Hardt and Lin consider the following question: given a simply connected domain ⊂ R2 and a map g : ∂ → S 1 of nonzero degree, what can be said about the limiting behavior of maps 1, p
u p ∈ Wg (, S 1 ) := {u ∈ W 1, p (, S 1 ) | u p |∂ = g} minimizing the p-energy 1, p |du p | p = min |du| p | u ∈ Wg (, S 1 )
Communicated by A. Neves.
B 1
Daniel Stern [email protected] Department of Mathematics, Princeton University, Princeton, NJ 08544, USA 0123456789().: V,-vol
123
187
Page 2 of 47
D. Stern
as p ∈ (1, 2) approaches 2 from below? They succeed in showing—among other things— that away from a collection A of |deg(g)| singularities, a subsequence u p j converges strongly 1 ( \ A, S 1 ), and the measures to a harmonic map v ∈ Cloc μ j = (2 − p j )|du j | p j (z)dz converge to the sum a∈A 2πδa of Dirac masses on A [19]. Moreover, the singular set A = {a1 , . . . , a| deg(g)| } minimizes a certain “renormalized energy” function Wg : | deg(g)| → [0, ∞] associated to g, providing a strong constraint on the location of the singularities. In particular, though the homotopically nontrivial boundary map g admits no extension to an S 1 -valued map of finite Dirichlet energy—i.e, Wg1,2 (, S 1 ) = ∅—the limit of the p-energy minimizers as p ↑ 2 provides us with a natural candidate for the optimal harmonic extension of g to an S 1 -valued map on . The results of [19] were inspired in large part by the analysis of Bethuel, Brezis, and Hélein—contained in the influential monograph [4]—of the asymptotics for minimizers u of the Ginzburg–Landau functionals E : W 1,2 (, R2 ) → R, E (u) =
1 (1 − |u|2 )2 |du|2 + 2 4 2
as → 0, with the measures μ :=
|du (z)|2 dz 2| log |
taking on the role played by the measures (2 − p)|du| p (z)dz in the setting of [19]. In recent decades, the asymptotics for critical points of the Ginzburg–Landau functional
Data Loading...
