Asymptotic Behavior for Minimizers of a P-energy Functional Associated with P-harmonic Maps

The author studies the asymptotic behavior of minimizers u ε of a p-energy functional with penalization as ε → 0. Several kinds of convergence for the minimizer to the p-harmonic map are presented under different assumptions.


Introduction
Let G ⊂ R 2 be a bounded and simply connected domain with smooth boundary ∂G, and B 1 = {x ∈ R 2 or the complex plane C; x 2  1 + x 2 2 < 1}.Denote S 1 = {x ∈ R 3 ; x 2 1 + x 2 2 = 1, x 3 = 0} and S 2 = {x ∈ R 3 ; x 2 1 + x 2 2 + x 2 3 = 1}.The vector value function can be denoted as u = (u 1 , u 2 , u 3 ) = (u , u 3 ).Let g = (g , 0) be a smooth map from ∂G into S 1 .Recall that the energy functional with a small parameter ε > 0 was introduced in the study of some simplified model of high-energy physics, which controls the statics of planner ferromagnets and antiferromagnets (see [9] and [12]).The asymptotic behavior of minimizers of E ε (u) had been studied by Fengbo Hang and Fanghua Lin in [7].When the term and S 2 replaced by R 2 , the problem becomes the simplified model of the Ginzburg-Landau theory for superconductors and was well studied in many papers such as [1][2] and [13].These works show that the properties of harmonic map with S 1 -value can be studied via researching the minimizers of the functional with some penalization terms.Indeed, Y.Chen and M.Struwe used the penalty method to establish the global existence of partial regular weak solutions of the harmonic map flow (see [4] and [6]).M.Misawa studied the p-harmonic maps by using the same idea of the penalty method in EJQTDE, 2004 No. 16, p. 1 [11].Now, the functional which equipped with the penalization 1 2ε p G u 2 3 dx, will be considered in this paper.From the direct method in the calculus of variations, it is easy to see that the functional achieves its minimum in the function class W 1,p g (G, S 2 ).Without loss of generality, we assume u 3 ≥ 0, otherwise we may consider |u 3 | in view of the expression of the functional.We will research the asymptotic properties of minimizers of this p-energy functional on W 1,p g (G, S 2 ) as ε → 0, and shall prove the limit of the minimizers is the p-harmonic map.
where u p is the minimizer of G |∇u| p dx in W 1,p g (G, ∂B 1 ).
Remark.When p = 2, [7] shows that if deg(g , ∂G) = 0, the minimizer of , where u 2 is the energy minimizer, i.e., it is the minimizer of G |∇u| 2 dx in H 1 g (G, ∂B 1 ).When p > 2, there may be several minimizers of ).The author proved that there exists a minimizer, which is called the regularized minimizer, is just (u p , 0), where u p is the minimizer of G |∇u| p dx in W 1,p g (G, ∂B 1 ).For the other minimizers, we only deduced the result as Theorem 1.1.
Comparing with the assumption of Theorem 1.1, we will consider the problem under some weaker conditions.Then we have for some subdomain K ⊆ G. Then there exists a subsequence u ε k of u ε such that as k → ∞, where u p is a critical point of The convergent rate of |u ε | → 1 and u 3 → 0 will be concerned with as ε → 0.
Theorem 1.3 Let u ε be a minimizer of E ε (u, G) on W 1,p g (G, S 2 ).If (1.1) holds, then there exists a positive constant C, such that as ε → 0, 2 Proof of Theorem 1.1 In this section, we always assume deg(g , ∂G) = 0.By the argument of the weak low semi-continuity, it is easy to deduce the strong convergence in W 1,p sense for some subsequence of the minimizer u ε .To improve the conclusion of the convergence for all u ε , we need to research the limit function: p-harmonic map.
From deg(g , ∂G) = 0 and the smoothness of ∂G and g, we see that there is a smooth function φ 0 : ∂G → R such that g = e iφ0 , on ∂G. (2.1) Assume Φ is the unique weak solution of (2.2) and (2.3).Set ) and Φ is the weak solution of (2.2) and (2.3), we obtain The solution is called p-energy minimizer.
EJQTDE, 2004 No. 16, p. 4 Proof.The weakly low semi-continuity of G |∇u| p dx is well-known.On the other hand, if taking a minimizing sequence u k of G |∇u| p dx in W 1,p g (G, ∂B 1 ), then there is a subsequence of u k , which is still denoted u k itself, such that as k → ∞, u k converges to u 0 weakly in W 1,p (G, C).Noting that W 1,p g (G, ∂B 1 ) is the weakly closed subset of W 1,p (G, C) since it is the convex closed subset, we see that u 0 ∈ W 1,p g (G, ∂B 1 ).Thus, if denote This means u 0 is the solution of (2.8).
Obviously, the p-energy minimizer is the p-harmonic map.
Proposition 2.4 The p-harmonic map is unique in W 1,p g (G, ∂B 1 ).
Proof.It follows that u p = e iΦ is a p-harmonic map from Proposition 2.2.If u is also a p-harmonic map in W 1,p g (G, ∂B 1 ), then from deg(g , ∂G) = 0 and using the results in [3], we know that there is Φ Substituting these into (2.6), we see that Φ 0 is a weak solution of (2.2) and (2.3).Proposition 2.1 leads to Φ 0 = Φ, which implies u = u p .Now, we conclude that u 0 in Proposition 2.3 is just the p-harmonic map u p .Furthermore, the p-energy minimizer is also unique in W 1,p g (G, ∂B 1 ).
Proof of Theorem 1.1.Noticing that u ε is the minimizer, we have ) Obviously, (2.11) and (2.13) lead to u * ∈ W 1,p g (G, S 1 ).Applying (2.12) and the weak low semi-continuity of G |∇u| p dx, we have On the other hand, (2.9) implies This means that u * is also a p-energy minimizer.Noting the uniqueness we see Combining this with (2.12) yields lim In addition, (2.13) implies that as ε → 0, Then lim Noticing the uniqueness of (u p , 0), we see the convergence above also holds for all u ε .EJQTDE, 2004 No. 16, p. 6 3 Proof of Theorem 1.2 In this section, we always assume that u ε is the critical point of the functional, and E ε (u ε , K) ≤ C for some subdomain K ⊆ G, where C is independent of ε.
The assumption is weaker than that of Theorem 1.1.So, all the results in this section will be derived in the weak sense.
The method in the calculus of variations shows that the minimizer where e 3 = (0, 0, 1).Namely, for any where C is independent of ε.Combining the fact |u ε | = 1 a.e. on G with (3.3) we know that there exist u p ∈ W 1,p (K, ∂B 1 ) and a subsequence u ε k of u ε , such that as ε k → 0, for some α ∈ (0, 1 − 2 p ).In the following we will prove that u p is a weak solution of (2.5). Let 2) and take ψ = (0, 0, φ).Thus Applying (3.3) we can derive that  ) we obtain that for any q ∈ (1, p), as ) is an arbitrary disc in K, we can see that as ε k → 0, for any ξ ∈ C ∞ 0 (B, R 3 ) there holds we have that for m, j ∈ {1, 2}, and m = j, One equation subtracts the other one, then 0 = where u ∧ ∇u = u 1 ∇u 2 − u 2 ∇u 1 .On the other hand, since we obtain that as  It shows that u p is a weak solution of (2.5).(1.2) is completed.

A Preliminary Proposition
To present the convergent rate of |u ε | → 1 and u ε3 → 0 in W 1,p sense when ε → 0, we need the following Then there exists a positive constant C which is independent of ε ∈ (0, 1), such that (4.3) Applying (1.1) and the integral mean value theorem, we know that there is a constant r ∈ (2R, 3R) such that Consider the functional where B = B(x, r).It is easy to prove that the minimizer ρ 1 of E(ρ, B) on ) where . Thus, by noting that ρ 1 is a minimizer, and applying (1.1) we see easily that Multiplying (4.5) by (ν • ∇ρ), where ρ denotes ρ 1 , and integrating over B, we have where ν denotes the unit vector on B, and it equals to the unit outside norm vector on ∂B.