Electronic Journal of Qualitative Theory of Differential Equations 2004, No. 16, 1-17; http://www.math.u-szeged.hu/ejqtde/

Asymptotic behavior for minimizers of a

p-energy functional associated with p-harmonic maps

Yutian Lei

Depart. of Math., Nanjing Normal University, Nanjing 210097, P.R.China E-mail: leiyutian@njnu.edu.cn

Abstract 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. Keywords: p-energy functional, p-energy minimizer, p-harmonic map MSC 35B25, 35J70, 49K20, 58G18

1 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_{1}^{2} + x_{2}^{2} < 1}. Denote S^{1} = {x ∈ R^{3}; x_{1}^{2} + x_{2}^{2} = 1, x_{3} = 0} and S^{2} = {x ∈ R^{3}; x_{1}^{2} + x_{2}^{2} + x_{3}^{2} = 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

E_{ε}(u) = \frac{1}{2} ∫_{G} |∇u|^{2}dx + \frac{1}{2ε^{2}} ∫_{G} u_{3}^{2}dx

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 \frac{u_{3}^{2}}{2ε^{2}} replaced by \frac{(1−|u|^{2})^{2}}{4ε^{2}} 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 [11]. Now, the functional

E_{ε}(u, G) = \frac{1}{p} ∫_{G} |∇u|^{p}dx + \frac{1}{2ε^{p}} ∫_{G} u_{3}^{2}dx, p > 2,

which equipped with the penalization \frac{1}{2ε^{p}} ∫_{G} u_{3}^{2}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_{g}^{1,p}(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_{g}^{1,p}(G, S^{2}) as ε → 0, and shall prove the limit of the minimizers is the p-harmonic map.

Theorem 1.1 Let u_{ε} be a minimizer of E_{ε}(u, G) on W_{g}^{1,p}(G, S^{2}). Assume deg(g′, ∂G) = 0. Then

lim_{ε→0} u_{ε} = (u_{p}, 0), in W^{1,p}(G, S^{2}),

where u_{p} is the minimizer of ∫_{G} |∇u|^{p}dx in W_{g}^{1,p}(G, ∂B_{1}).

Remark. When p = 2, [7] shows that if deg(g′, ∂G) = 0, the minimizer of E_{ε}(u) in H_{g}^{1}(G, S^{2}) is just (u_{2}, 0), where u_{2} is the energy minimizer, i.e., it is the minimizer of ∫_{G} |∇u|^{2}dx in H_{g}^{1}(G, ∂B_{1}). When p > 2, there may be several minimizers of E_{ε}(u, G) in W_{g}^{1,p}(G, S^{2}). 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_{g}^{1,p}(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

Theorem 1.2 Assume u_{ε} is a critical point of E_{ε}(u, G) on W_{g}^{1,p}(G, S^{2}). If

E_{ε}(u_{ε}, K) ≤ C  (1.1)

for some subdomain K ⊆ G. Then there exists a subsequence u_{ε_{k}} of u_{ε} such that as k → ∞,

u_{ε_{k}} → (u_{p}, 0), weakly in W^{1,p}(K, R^{3}),  (1.2)

where u_{p} is a critical point of ∫_{K} |∇u|^{p}dx in W^{1,p}(K, ∂B_{1}), which is named p-harmonic map on K. Moreover, for any ζ ∈ C_{0}^{∞}(K), when ε → 0,

∫_{K} |∇u_{ε_{k}}|^{p}ζdx → ∫_{K} |∇u_{p}|^{p}ζdx,  (1.3)

\frac{1}{ε_{k}^{p}} ∫_{K} u_{ε_{k}3}ζdx → 0.  (1.4)

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_{g}^{1,p}(G, S^{2}). If (1.1) holds, then there exists a positive constant C, such that as ε → 0,

∫_{K} |∇|u′_{ε}||^{p}dx + ∫_{K} |∇u_{ε3}|^{p}dx + \frac{1}{ε^{p}} ∫_{K} u_{ε3}^{2}dx ≤ Cε^{β},

where β = 1 − \frac{2}{p} when p ∈ (2, p_{0}]; β = \frac{2p}{p^{2}−2} when p > p_{0}. Here p_{0} ∈ (4, 5) is a constant satisfying p^{3} − 4p^{2} − 2p + 4 = 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^{iϕ_{0}}, on ∂G.  (2.1)

Consider the Dirichlet problem

−div(|∇Φ|^{p−2}∇Φ) = 0, in G,  (2.2)
Φ|_{∂G} = ϕ_{0}.  (2.3)

Proposition 2.1 There exists the unique weak solution Φ of (2.2) and (2.3) in W^{1,p}(G, R). Namely, for any ϕ ∈ W_{0}^{1,p}(G, R), there is the unique Φ satisfies

∫_{G} |∇Φ|^{p−2}∇Φ∇ϕdx = 0  (2.4)

Proof. By using the method in the calculus of variations, we can see the existence for the weak solution of (2.2) and (2.3).

If both Φ_{1} and Φ_{2} are weak solutions of (2.2) and (2.3), then, by taking the test function ϕ = Φ_{1} − Φ_{2} in (2.4), there holds

∫_{G}(|∇Φ_{1}|^{p−2}∇Φ_{1} − |∇Φ_{2}|^{p−2}∇Φ_{2})∇(Φ_{1} − Φ_{2})dx = 0.

In view of Lemma 1.2 in [5] we have

∫_{G} |∇(Φ_{1} − Φ_{2})|^{p}dx ≤ 0.

Hence, Φ_{1}−Φ_{2} = Const. on \overline{G}. Noting the boundary condition, we see Φ_{1}−Φ_{2} = 0 on \overline{G}. Proposition is proved.

Recall that u ∈ W_{g}^{1,p}(G, ∂B_{1}) is named p-harmonic map, if it is the critical point of ∫_{G} |∇u|^{p}dx. Namely, it is the weak solution of

−div(|∇u|^{p−2}∇u) = u|∇u|^{p}  (2.5)

on G, or for any ϕ ∈ C_{0}^{∞}(G, R^{2} or C), it satisfies

∫_{G} |∇u|^{p−2}∇u∇ϕdx = ∫_{G} u|∇u|^{p}ϕdx.  (2.6)

Assume Φ is the unique weak solution of (2.2) and (2.3). Set

u_{p} = e^{iΦ}, on \overline{G}.  (2.7)

Proposition 2.2 u_{p} defined in (2.7) is a p-harmonic map on G.

Proof. Obviously, u_{p} ∈ W_{g}^{1,p}(G, ∂B_{1}) since Φ ∈ W_{ϕ_{0}}^{1,p}(G, R). We only need to prove that u_{p} satisfies (2.6) for any ϕ ∈ C_{0}^{∞}(G, C). In fact,

∫_{G}(|∇u_{p}|^{p−2}∇u_{p}∇ϕ − u_{p}ϕ|∇u_{p}|^{p})dx

= i ∫_{G} |∇Φ|^{p−2}∇Φ(e^{iΦ}∇ϕ + ie^{iΦ}∇Φϕ)dx = i ∫_{G} |∇Φ|^{p−2}∇Φ∇(e^{iΦ}ϕ)dx

for any ϕ ∈ C_{0}^{∞}(G, C). Noting e^{iΦ}ϕ ∈ W_{0}^{1,p}(G, C) and Φ is the weak solution of (2.2) and (2.3), we obtain

∫_{G} |∇u_{p}|^{p−2}∇u_{p}∇ϕdx − ∫_{G} u_{p}ϕ|∇u_{p}|^{p}dx = 0

for any ϕ ∈ C_{0}^{∞}(G, C). Proposition is proved.

Since W_{g}^{1,p}(G, ∂B_{1}) ≠ ∅ when deg(g′, ∂G) = 0, we may consider the minimization problem

Min{∫_{G} |∇u|^{p}dx; u ∈ W_{g}^{1,p}(G, ∂B_{1})}  (2.8)

The solution is called p-energy minimizer.

Proposition 2.3 The solution of (2.8) exists.

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_{g}^{1,p}(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_{g}^{1,p}(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_{g}^{1,p}(G, ∂B_{1}). Thus, if denote

α = Inf{∫_{G} |∇u|^{p}dx; u ∈ W_{g}^{1,p}(G, ∂B_{1})},

then

α ≤ ∫_{G} |∇u_{0}|^{p}dx ≤ \underline{lim}_{k→∞} ∫_{G} |∇u_{k}|^{p}dx ≤ α.

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_{g}^{1,p}(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_{g}^{1,p}(G, ∂B_{1}), then from deg(g′, ∂G) = 0 and using the results in [3], we know that there is Φ_{0} ∈ W^{1,p}(G, R) such that

u = e^{iΦ_{0}}, on \overline{G},
Φ_{0} = ϕ_{0}, on ∂G.

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_{g}^{1,p}(G, ∂B_{1}).

Proof of Theorem 1.1. Noticing that u_{ε} is the minimizer, we have

E_{ε}(u_{ε}, G) ≤ E_{ε}((u_{p}, 0), G) ≤ C  (2.9)

with C > 0 independent of ε. This means

∫_{G} |∇u_{ε}|^{p}dx ≤ C,  (2.10)

∫_{G} u_{ε3}^{2}dx ≤ Cε^{p}.  (2.11)

Using (2.10), |u_{ε}| = 1 and the embedding theorem, we see that there exists a subsequence u_{ε_{k}} of u_{ε} and u_{∗} ∈ W^{1,p}(G, R^{3}), such that as ε_{k} → 0,

u_{ε_{k}} → u_{∗}, weakly in W^{1,p}(G, S^{2}),  (2.12)
u_{ε_{k}} → u_{∗}, in C^{α}(\overline{G}, S^{2}), α ∈ (0, 1 − 2/p).  (2.13)

Obviously, (2.11) and (2.13) lead to u_{∗} ∈ W_{g}^{1,p}(G, S^{1}).

Applying (2.12) and the weak low semi-continuity of ∫_{G} |∇u|^{p}dx, we have

∫_{G} |∇u_{∗}|^{p}dx ≤ \underline{lim}_{ε_{k}→0} ∫_{G} |∇u_{ε_{k}}|^{p}dx.

On the other hand, (2.9) implies

∫_{G} |∇u_{ε_{k}}|^{p}dx ≤ ∫_{G} |∇(u_{p}, 0)|^{p}dx,

hence,

∫_{G} |∇u′_{∗}|^{p}dx ≤ ∫_{G} |∇u_{p}|^{p}dx.

This means that u′_{∗} is also a p-energy minimizer. Noting the uniqueness we see u_{∗} = u_{p}. Thus

∫_{G} |∇u_{p}|^{p}dx ≤ \underline{lim}_{ε_{k}→0} ∫_{G} |∇u_{ε_{k}}|^{p}dx ≤ \overline{lim}_{ε_{k}→0} ∫_{G} |∇u_{ε_{k}}|^{p}dx ≤ ∫_{G} |∇u_{p}|^{p}dx.

When ε_{k} → 0,

∫_{G} |∇u_{ε_{k}}|^{p} → ∫_{G} |∇u_{p}|^{p}.

Combining this with (2.12) yields

lim_{k→∞} ∇u_{ε_{k}} = ∇(u_{p}, 0), in L^{p}(G, S^{2}).

In addition, (2.13) implies that as ε → 0,

u_{ε_{k}} → (u_{p}, 0), in L^{p}(G, S^{2}).

Then

lim_{k→∞} u_{ε_{k}} = (u_{p}, 0), in W^{1,p}(G, S^{2}).

Noticing the uniqueness of (u_{p}, 0), we see the convergence above also holds for all u_{ε}.

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 u_{ε} ∈ W_{g}^{1,p}(G, S^{2}) is a weak solution of

−div(|∇u|^{p−2}∇u) = u|∇u|^{p} + \frac{1}{ε^{p}} (uu_{3}^{2} − u_{3}e_{3}), on G,  (3.1)

where e_{3} = (0, 0, 1). Namely, for any ψ ∈ W_{0}^{1,p}(G, R^{3}), u_{ε} satisfies

∫_{G} |∇u|^{p−2}∇u∇ψdx = ∫_{G} uψ|∇u|^{p}dx + \frac{1}{ε^{p}} ∫_{G} ψ(uu_{3}^{2} − u_{3}e_{3})dx.  (3.2)

Proof of (1.2). E_{ε}(u_{ε}, K) ≤ C means

∫_{K} |∇u_{ε}|^{p}dx ≤ C,  (3.3)

∫_{K} u_{ε3}^{2}dx ≤ Cε^{p},  (3.4)

where C is independent of ε. Combining the fact |u_{ε}| = 1 a.e. on \overline{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,

u_{ε_{k}} → (u_{p}, 0), weakly in W^{1,p}(K),  (3.5)
u_{ε_{k}} → (u_{p}, 0), in C^{α}(\overline{K}),  (3.6)

for some α ∈ (0, 1 − \frac{2}{p}). In the following we will prove that u_{p} is a weak solution of (2.5).

Let B = B(x, 3R) ⊂⊂ K. ϕ ∈ C_{0}^{∞}(B(x, 3R); [0, 1]), ϕ = 1 on B(x, R), ϕ = 0 on B \ B(x, 2R) and |∇ϕ| ≤ C, where C is independent of ε. Denote u = u_{ε_{k}} in (3.2) and take ψ = (0, 0, ϕ). Thus

∫_{B} |∇u|^{p−2}∇u_{3}∇ϕdx + \frac{1}{ε_{k}^{p}} ∫_{B} |u′|^{2}ϕu_{3}dx = ∫_{B} u_{3}ϕ|∇u|^{p}dx.

Applying (3.3) we can derive that

\frac{1}{ε_{k}^{p}} ∫_{B} |u′|^{2}ϕ|u_{3}|dx ≤ ∫_{B} |∇u|^{p}ϕdx + ∫_{B} |∇u|^{p−1}|∇ϕ|dx ≤ C.  (3.7)

From (3.6) it follows |u′| ≥ 1/2 when ε_{k} is sufficiently small. Noting ϕ = 1 on B(x, R), we have

\frac{1}{ε_{k}^{p}} ∫_{B(x,R)} |u_{3}|dx ≤ C.  (3.8)

Taking \frac{1}{k} = ε_{k}, F_{k} = \frac{1}{ε_{k}^{p}} (u_{ε_{k}}u_{ε_{k}3}^{2} − u_{ε_{k}3}e_{3}) in Lemma 3.11 of [8], noting |F_{k}| = \frac{1}{ε_{k}^{p}} |u_{3}||u′| and applying (3.5) and (3.8) we obtain that for any q ∈ (1, p), as ε_{k} → 0, ∇u_{ε_{k}} → ∇u_{p}, in L^{q}(B(x, R)). Since B(x, R) is an arbitrary disc in K, we can see that as ε_{k} → 0, for any ξ ∈ C_{0}^{∞}(B, R^{3}) there holds

∫_{B} |∇u_{ε_{k}}|^{p−2}∇u_{ε_{k}}∇ξdx → ∫_{B} |∇u_{p}|^{p−2}∇u_{p}∇ξdx.  (3.9)

Now, denote u′ = u′_{ε_{k}} = (u_{1}, u_{2}). Taking ψ = (u_{2}, 0, 0)ζ and ψ = (0, u_{1}, 0)ζ in (3.2), respectively, where ζ ∈ C_{0}^{∞}(B, R), we have that for m, j ∈ {1, 2}, and m ≠ j,

\begin{matrix}\frac{1}{ε_{k}^{p}} ∫_{B} u_{3}^{2}u_{m}u_{j}ζdx + ∫_{B} u_{m}u_{j}ζ|∇u|^{p}dx \\ = ∫_{B} |∇u|^{p−2}∇u_{m}∇u_{j}ζdx + ∫_{B} u_{j}|∇u|^{p−2}∇u_{m}∇ζdx. \end{matrix}

One equation subtracts the other one, then

0 = ∫_{B} |∇u|^{p−2}(u ∧ ∇u)∇ζdx,  (3.10)

where u ∧ ∇u = u_{1}∇u_{2} − u_{2}∇u_{1}. On the other hand, since

∫_{B} u_{2}|∇u|^{p−2}∇u_{1}∇ζdx − ∫_{B} u_{p2}|∇u_{p}|^{p−2}∇u_{p1}∇ζdx
= ∫_{B}(|∇u|^{p−2}∇u_{1} − |∇u_{p}|^{p−2}∇u_{p1})u_{p2}∇ζdx
+ ∫_{B} |∇u|^{p−2}∇u_{1}∇ζ(u_{2} − u_{p2})dx,

we obtain that as ε_{k} → 0,

∫_{B} u_{2}|∇u|^{p−2}∇u_{1}∇ζdx → ∫_{B} u_{p2}|∇u_{p}|^{p−2}∇u_{p1}∇ζdx  (3.11)

by using (3.3)(3.6) and (3.9). Similarly, we may also get that

lim_{ε→0} ∫_{B} u_{1}|∇u|^{p−2}∇u_{2}∇ζdx = ∫_{B} u_{p1}|∇u_{p}|^{p−2}∇u_{p2}∇ζdx.  (3.12)

(3.12) subtracts (3.11), then

lim_{ε→0} ∫_{B} |∇u|^{p−2}(u ∧ ∇u)∇ζdx = ∫_{B} |∇u_{p}|^{p−2}(u_{p} ∧ ∇u_{p})∇ζdx.

Combining this with (3.10), we have

∫_{B} |∇u_{p}|^{p−2}(u_{p} ∧ ∇u_{p})∇ζdx = 0.  (3.13)

Let u_{∗} = u_{p1} + iu_{p2} : B → C. Thus

|∇u_{∗}|^{2} = |∇u_{p}|^{2}.  (3.14)

It is easy to see that \overline{u_{∗}}∇u_{∗} = ∇(|u_{∗}|^{2}) + (u_{∗} ∧ ∇u_{∗})i = 0 + (u_{∗} ∧ ∇u_{∗})i since |u_{∗}|^{2} = |u_{p1}|^{2} + |u_{p2}|^{2} = 1. Substituting this into (3.13) yields

−i ∫_{B} |∇u_{∗}|^{p−2}\overline{u_{∗}}∇u_{∗}∇ζdx = 0

for any ζ ∈ C_{0}^{∞}(B, R). Taking ζ = Re(u_{∗}ϕ_{j}) and ζ = Im(u_{∗}ϕ_{j}) (j = 1, 2), respectively, where ϕ = (ϕ_{1}, ϕ_{2}) ∈ C_{0}^{∞}(B, R^{2}), we can see that

∫_{B} |∇u_{∗}|^{p−2}\overline{u_{∗}}∇u_{∗}∇Re(u_{∗}ϕ)dx + i ∫_{B} |∇u_{∗}|^{p−2}\overline{u_{∗}}∇u_{∗}∇Im(u_{∗}ϕ)dx = 0.

Namely

0 = ∫_{G} |∇u_{∗}|^{p−2}\overline{u_{∗}}∇u_{∗}∇(u_{∗}ϕ)dx.

Noting \overline{u_{∗}}∇u_{∗} = −u_{∗}∇\overline{u_{∗}}, we obtain

\begin{aligned}0 & = ∫_{B} |∇u_{∗}|^{p−2}∇u_{∗}∇ϕdx − ∫_{B} |∇u_{∗}|^{p−2}u_{∗}∇\overline{u_{∗}}∇u_{∗}ϕdx \\ & = ∫_{B} |∇u_{∗}|^{p−2}∇u_{∗}∇ϕdx − ∫_{B} |∇u_{∗}|^{p}u_{∗}ϕdx := J \end{aligned}

By using (3.14) and Re(J) = 0, Im(J) = 0, we have

∫_{B} |∇u_{p}|^{p−2}∇u_{p1}∇ϕdx = ∫_{B} |∇u_{p}|^{p}u_{p1}ϕdx  (3.15)

and

∫_{B} |∇u_{p}|^{p−2}∇u_{p2}∇ϕdx = ∫_{B} |∇u_{p}|^{p}u_{p2}ϕdx.

Combining this with (3.15) yields that for any ϕ ∈ C_{0}^{∞}(B, R^{3}),

∫_{B} |∇u_{p}|^{p−2}∇u_{p}∇ϕdx = ∫_{B} |∇u_{p}|^{p}u_{p}ϕdx.

It shows that u_{p} is a weak solution of (2.5). (1.2) is completed.

Proof of (1.3). For simplification, denote ε_{k} = ε. From (3.3) and (3.6) it is deduced that as ε → 0,

| ∫_{K} u_{3}^{2}ζ|∇u|^{p}dx| ≤ sup_{K}(1 − |u′|^{2}) · ∫_{K} |∇u|^{p}dx → 0,  (3.16)

\begin{matrix}| ∫_{K} u′u_{p}ζ|∇u|^{p}dx − ∫_{K} ζ|∇u|^{p}dx| = | ∫_{K}(u′u_{p} − u_{p}u_{p})ζ|∇u|^{p}dx| \\ ≤ sup_{K} |u′ − u_{p}| · | ∫_{K} u_{p}|∇u|^{p}dx| → 0, \end{matrix} (3.17)

and

∫_{K}(u − (u_{p}, 0))ζ|∇u|^{p}dx ≤ sup_{K} |u − (u_{p}, 0)| · | ∫_{K} u_{p}|∇u|^{p}dx| → 0. (3.18)

Similarly, (3.4) and (3.6) imply that as ε → 0,

|\frac{1}{ε^{p}} ∫_{K} u_{3}^{2}ζdx − \frac{1}{ε^{p}} ∫_{K} u_{3}^{2}ζ(1 − u_{3}^{2})dx| ≤ sup_{K} |1 − |u′|^{2}| · \frac{1}{ε^{p}} | ∫_{K} u_{3}^{2}dx| → 0 (3.19)

and

|\frac{1}{ε^{p}} ∫_{K} u_{p}ζu′u_{3}^{2}dx − \frac{1}{ε^{p}} ∫_{K} ζu_{3}^{2}dx| ≤ sup_{K} |u′ − u_{p}| · \frac{1}{ε^{p}} | ∫_{K} u_{p}u_{3}^{2}dx| → 0. (3.20)

Letting ε → 0 in (3.2) we have

\begin{matrix}lim_{ε→0}[∫_{K} uψ|∇u|^{p}dx + \frac{1}{ε^{p}} ∫_{K} ψ(uu_{3}^{2} − u_{3}e_{3})dx] \\ = ∫_{K} |∇u_{p}|^{p−2}∇(u_{p}, 0)∇ψdx = ∫_{G}(u_{p}, 0)ψ|∇u_{p}|^{p}dx. \end{matrix}  (3.21)

Take ψ = (0, 0, u_{3}ζ) where ζ ∈ C_{0}^{∞}(K) we have

lim_{ε→0}[∫_{K} u_{3}^{2}ζ|∇u|^{p}dx + \frac{1}{ε^{p}} ∫_{K} u_{3}^{2}ζ(u_{3}^{2} − 1)dx] = 0.

Combining this with (3.16) we derive

lim_{ε→0} \frac{1}{ε^{p}} ∫_{K} u_{3}^{2}ζ(u_{3}^{2} − 1)dx = 0.

Substituting this into (3.19) yields

lim_{ε→0} \frac{1}{ε^{p}} ∫_{K} u_{3}^{2}ζdx = 0.  (3.22)

Hence, as ε → 0,

\frac{1}{ε^{p}} | ∫_{K} uu_{3}^{2}ζdx| ≤ \frac{1}{ε^{p}} ∫_{K} u_{3}^{2}ζdx → 0.

Thus, for any ψ ∈ W_{0}^{1,p}(K, R^{3}), there holds

lim_{ε→0} \frac{1}{ε^{p}} ∫_{K} uu_{3}^{2}ψdx = 0.  (3.23)

In addition, substituting (3.22) into (3.20) leads to

lim_{ε→0} \frac{1}{ε^{p}} ∫_{K} u_{p}ζu′u_{3}^{2}dx = 0.  (3.24)

Take ψ = (u_{p}ζ, 0) in (3.21) we have

lim_{ε→0}[∫_{K} u′u_{p}ζ|∇u|^{p}dx + \frac{1}{ε^{p}} ∫_{K} u_{p}ζu′u_{3}^{2}dx] = ∫_{K} |∇u_{p}|^{p}ζdx,

which, together with (3.24), implies

lim_{ε→0} ∫_{K} u′u_{p}ζ|∇u|^{p}dx = ∫_{K} |∇u_{p}|^{p}ζdx.

Combining this with (3.17) we can see (1.3) at last.

Proof of (1.4). Obviously, (3.18) and (1.3) show that as ε → 0,

\begin{matrix}| ∫_{K} u|∇u|^{p}ψdx − ∫_{K}(u_{p}, 0)|∇u_{p}|^{p}ψdx| \\ ≤ | ∫_{K}(u − (u_{p}, 0))|∇u|^{p}ψdx| + | ∫_{K}(u_{p}, 0)(|∇u|^{p} − |∇u_{p}|^{p})ψdx| → 0. \end{matrix}

Substituting this and (3.23) into (3.21), we see that the left hand side of (3.21) becomes

\begin{matrix}lim_{ε→0}[∫_{K} uψ|∇u|^{p}dx + \frac{1}{ε^{p}} ∫_{K} ψ(uu_{3}^{2} − u_{3}e_{3})dx] \\ = ∫_{K}(u_{p}, 0)|∇u_{p}|^{p}ψdx − lim_{ε→0} \frac{1}{ε^{p}} ∫_{K} ψu_{3}e_{3}dx. \end{matrix}

Comparing this with the right hand side of (3.21), we have

lim_{ε→0} \frac{1}{ε^{p}} ∫_{K} ψu_{3}e_{3}dx = 0.

This is (1.4). Theorem 1.2 is proved.

4 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

Proposition 4.1 Assume u_{ε} is a minimizer of E_{ε}(u, G) on W. If E_{ε}(u_{ε}, K) ≤ C for some domain K ⊆ G. Then there exists a positive constant C which is independent of ε ∈ (0, 1), such that

\frac{1}{p} ∫_{K} |∇u_{ε}|^{p}dx + \frac{1}{ε^{p}} ∫_{K} u_{ε3}^{2}dx ≤ Cε^{2/p} + \frac{1}{p} ∫_{K} |∇\frac{u′_{ε}}{|u′_{ε}|}|^{p}dx.  (4.1)

Proof. Denote w = \frac{u′_{ε}}{|u′_{ε}|}. Choose R > 0 sufficiently small such that \overline{B(x, 3R)} ⊂ K. It follows from (3.6) that

|u′_{ε}| ≥ 1/2  (4.2)

on B(x, 3R) as ε sufficiently small. This and (3.3) imply

∫_{B(x,3R)} |∇w|^{p}dx ≤ 2^{p} ∫_{B(x,3R)} |u′_{ε}|^{p}|∇w|^{p}dx ≤ C ∫_{B(x,3R)} |∇u_{ε}|^{p}dx ≤ C.

(4.3)

Applying (1.1) and the integral mean value theorem, we know that there is a constant r ∈ (2R, 3R) such that

\frac{1}{p} ∫_{∂B(x,r)} |∇u_{ε}|^{p}dx + \frac{1}{2ε^{p}} ∫_{∂B(x,r)} u_{ε3}^{2}dx = C_{0}(r)E_{ε}(u_{ε}, B_{3R} \ B_{2R}) ≤ C. (4.4)

Consider the functional

E(ρ, B) = \frac{1}{p} ∫_{B}(|∇ρ|^{2} + 1)^{p/2}dx + \frac{1}{2ε^{p}} ∫_{B}(1 − ρ)^{2}dx,

where B = B(x, r). It is easy to prove that the minimizer ρ_{1} of E(ρ, B) on W_{|u′_{ε}|}^{1,p}(B, R^{+} ∪ {0}) exists and solves

−div(v^{(p−2)/2}∇ρ) = \frac{1}{ε^{p}} (1 − ρ) on B,  (4.5)
ρ|_{∂B} = |u′_{ε}|,  (4.6)

where v = |∇ρ|^{2} + 1. Since 1/2 < |u′_{ε}| ≤ 1, it follows from the maximum principle that on \overline{B},

\frac{1}{2} < ρ_{1} ≤ 1.  (4.7)

Clearly, (1 − |u′|)^{2} ≤ (1 − |u′|^{2})^{2} = u_{3}^{4} ≤ u_{3}^{2}. Thus, by noting that ρ_{1} is a minimizer, and applying (1.1) we see easily that

E(ρ_{1}, B) ≤ E(|u′_{ε}|, B) ≤ CE_{ε}(u_{ε}, B) ≤ C.  (4.8)

Multiplying (4.5) by (ν · ∇ρ), where ρ denotes ρ_{1}, and integrating over B, we have

\begin{aligned}− ∫_{∂B} v^{(p−2)/2}(ν · ∇ρ)^{2}dξ & + ∫_{B} v^{(n−2)/2}∇ρ · ∇(ν · ∇ρ)dx \\ & = \frac{1}{ε^{p}} ∫_{B}(1 − ρ)(ν · ∇ρ)dx, \end{aligned}  (4.9)

where ν denotes the unit vector on \overline{B}, and it equals to the unit outside norm vector on ∂B.

Using (4.8) we obtain

| ∫_{B} v^{(p−2)/2}∇ρ · ∇(ν · ∇ρ)dx|
≤ C ∫_{B} v^{(n−2)/2}|∇ρ|^{2}dx + \frac{1}{2}| ∫_{B} v^{(p−2)/2}(ν · ∇v)dx|
≤ C + \frac{1}{p}| ∫_{B} ν · ∇(v^{n/2})dx| ≤ C + \frac{1}{p} ∫_{B} |div(v^{p/2}ν) − v^{p/2}divν|dx
≤ C + \frac{1}{p} ∫_{∂B} v^{p/2}dξ.

Combining (4.6), (4.4) and (4.8) we also have

\begin{matrix}|\frac{1}{ε^{p}} ∫_{B}(1 − ρ)(ν · ∇ρ)dx| ≤ \frac{1}{2ε^{p}} | ∫_{B}(1 − ρ)^{2}divνdx − ∫_{∂B}(1 − ρ)^{2}dξ| \\ ≤ \frac{1}{2ε^{p}} ∫_{B}(1 − ρ)^{2}|divν|dx + \frac{1}{2ε^{p}} ∫_{∂B}(1 − ρ)^{2}dξ ≤ C. \end{matrix}

Substituting these into (4.9) yields

| ∫_{∂B} v^{(p−2)/2}(ν · ∇ρ)^{2}dξ| ≤ C + \frac{1}{p} ∫_{∂B} v^{p/2}dξ.  (4.10)

Applying (4.6), (4.4) and (4.10), we obtain for any δ ∈ (0, 1),

∫_{∂B} v^{p/2}dξ = ∫_{∂B} v^{(p−2)/2}[1 + (τ · ∇ρ)^{2} + (ν · ∇ρ)^{2}]dξ
≤ ∫_{∂B} v^{(p−2)/2}dξ + ∫_{∂B} v^{(p−2)/2}(ν · ∇ρ)^{2}dξ
+(∫_{∂B} v^{p/2}dξ)^{(p−2)/p}(∫_{∂B}(τ · ∇|u′_{ε}|)^{p}dξ)^{2/p}
≤ C(δ) + (\frac{1}{p} + 2δ) ∫_{∂B} v^{p/2}dξ,

where τ denotes the unit tangent vector on ∂B. Hence it follows by choosing δ > 0 so small that

∫_{∂B} v^{p/2}dξ ≤ C.  (4.11)

Now we multiply both sides of (4.5) by (1 − ρ) and integrate over B. Then

∫_{B} v^{(p−2)/2}|∇ρ|^{2}dx + \frac{1}{ε^{p}} ∫_{B}(1 − ρ)^{2}dx = − ∫_{∂B} v^{(p−2)/2}(ν · ∇ρ)(1 − ρ)dξ.

From this, using (4.4), (4.6), (4.7) and (4.11) we obtain

\begin{matrix}E(ρ_{1}, B) ≤ C| ∫_{∂B} v^{(p−2)/2}(ν · ∇ρ)(1 − ρ)dξ| \\ ≤ C| ∫_{∂B} v^{p/2}dξ|^{(p−1)/p}| ∫_{∂B}(1 − ρ)^{2}dξ|^{1/p} \\ ≤ C| ∫_{∂B}(1 − |u′_{ε}|)^{2}dξ|^{1/p} ≤ Cε \end{matrix}  (4.12)

Since u_{ε} is a minimizer of E_{ε}(u, G), we have

E_{ε}(u_{ε}, G) ≤ E_{ε}(U, G),

where

U = (ρ_{1}w, \sqrt{1 − ρ_{1}^{2}}) on B; U = u_{ε} on G \ B.

Namely,

E_{ε}(u_{ε}, G) ≤ E_{ε}(ρ_{1}w, B) + E_{ε}(u_{ε}, G \ B).

Hence

\begin{matrix}E_{ε}(u_{ε}, B) ≤ E_{ε}(ρ_{1}w, B) \\ = \frac{1}{p} ∫_{B}(|∇ρ_{1}|^{2} + ρ_{1}^{2}|∇w|^{2})^{p/2}dx + \frac{1}{2ε^{p}} ∫_{B}(1 − ρ_{1}^{2})dx, \end{matrix}  (4.13)

where w = \frac{u′_{ε}}{|u′_{ε}|}. On one hand,

\begin{matrix}∫_{B}(|∇ρ_{1}|^{2} + ρ_{1}^{2}|∇w|^{2})^{p/2}dx − ∫_{B}(ρ_{1}^{2}|∇w|^{2})^{p/2}dx \\ = \frac{p}{2} ∫_{B} ∫_{0}^{1}[(|∇ρ_{1}|^{2} + ρ_{1}^{2}|∇w|^{2})^{(p−2)/2}s \\ +(ρ_{1}^{2}|∇w|^{2})^{(p−2)/2}(1 − s)]ds|∇ρ_{1}|^{2}dx \\ ≤ C ∫_{B}(|∇ρ_{1}|^{p} + |∇ρ_{1}|^{2}|∇w|^{p−2})dx. \end{matrix}  (4.14)

On the other hand, by using (4.12) and (4.3) we have

∫_{B} |∇ρ_{1}|^{2}|∇w|^{p−2}dx ≤ (∫_{B} |∇ρ_{1}|^{p}dx)^{2/p}·(∫_{B} |∇w|^{p}dx)^{(p−2)/p} ≤ Cε^{2/p}. (4.15)

Combining (4.13)-(4.15), we can derive

E_{ε}(u_{ε}, B) ≤ \frac{1}{p} ∫_{B} ρ_{1}^{p}|∇w|^{p}dx + Cε^{2/p}.

Thus (4.1) can be seen by noticing (4.7).

5 Proof of Theorem 1.3

Assume that u_{ε} is a minimizer, and B = B(x, r). By noting p > 2 and using Jensen’s inequality, we have

E_{ε}(u_{ε}, B) ≥ \frac{1}{p} ∫_{B} |∇h|^{p}dx + \frac{1}{p} ∫_{B} h^{p}|∇w|^{p}dx + \frac{1}{p} ∫_{B} |∇u_{3}|^{p}dx + \frac{1}{2ε^{p}} ∫_{B} u_{3}^{2}dx,

EJQTDE, 2004 No. 16, p. 14 where h = |u′_{ε}|. Thus, from (4.1) it follows that,

\begin{matrix}\frac{1}{p} ∫_{B}(|∇h|^{p} + |∇u_{3}|^{p})dx + \frac{1}{p} ∫_{B}(h^{p} − 1)|∇w|^{p}dx + \frac{1}{2ε^{p}} ∫_{B} u_{3}^{2}dx \\ ≤ E_{ε}(u_{ε}, B) − \frac{1}{p} ∫_{B} |∇w|^{p}dx ≤ Cε^{2/p}. \end{matrix}  (5.1)

Since |u_{3}| ≤ 1 and (1.1), we have

|u_{3}(x) − u_{3}(y)| ≤ C∥u_{3}∥_{W^{1,p}(K)}|x − y|^{1−2/p} ≤ C|x − y|^{1−2/p}, ∀x, y ∈ K. Hence, u_{3}^{2}(x) ≥ (|u_{3}(y)| − Cε^{1−2/p})^{2} when x ∈ B(y, ε). Substituting this into (3.4) we obtain

π(|u_{3}(y)| − Cε^{1−2/p})^{2}ε^{2} ≤ ∫_{B(y,ε)} u_{3}^{2}(x)dx ≤ Cε^{p}

for any y ∈ K. This implies

sup_{y∈K} |u_{3}(y)| ≤ Cε^{1−2/p}.

Thus, by using (4.2) and (3.3), we have that for any constant δ ∈ (0, 1),

\begin{aligned}\frac{1}{p} ∫_{B}(1 − h^{p})|∇w|^{p}dx & ≤ \frac{2^{p}}{p} ∫_{B}(1 − h^{p})h^{p}|∇w|^{p}dx \\ & ≤ C ∫_{B} u_{3}^{2}|∇u_{ε}|^{p}dx ≤ Cε^{1−2/p}. \end{aligned}  (5.2)

Substituting this into (5.1), we can derive

∫_{B} |∇h|^{p}dx + ∫_{B} |∇u_{3}|^{p}dx + \frac{1}{ε^{p}} ∫_{B} u_{3}^{2}dx ≤ C(ε^{1−2/p} + ε^{2/p}).  (5.3)

If p ≤ 4, then we have finished. If p > 4, we will prove Theorem 5.1 Let p_{0} ∈ (4, 5) satisfy p^{3} − 4p^{2} − 2p + 4 = 0. Then

∫_{B} |∇h|^{p}dx + ∫_{B} |∇u_{3}|^{p}dx + \frac{1}{ε^{p}} ∫_{B} u_{3}^{2}dx ≤ Cε^{1−2/p}, when p ∈ (4, p_{0}];

∫_{B} |∇h|^{p}dx + ∫_{B} |∇u_{3}|^{p}dx + \frac{1}{ε^{p}} ∫_{B} u_{3}^{2}dx ≤ Cε^{\frac{2p}{p^{2}−2}} , when p > p_{0}.

Proof. \underline{Step 1.} The idea of Proposition 4.1 is used. At first, from (5.3) it follows that

∫_{B} u_{3}^{2}dx ≤ Cε^{\frac{2}{p} +p}.

Using this and the integral mean theorem, we see that there exists r_{2} ∈ (2R, r) such that

∫_{∂B(x,r_{2})} u_{3}^{2}dx ≤ Cε^{\frac{2}{p} +p}.

Next, consider the minimizer ρ_{2} of the functional

E(ρ, B(x, r_{2})) = \frac{1}{p} ∫_{B(x,r_{2})}(|∇ρ|^{2} + 1)^{p/2}dx + \frac{1}{2ε^{p}} ∫_{B(x,r_{2})}(1 − ρ)^{2}dx,

in W_{|u′_{ε}|}^{1,p}(B(x, r_{2}), R^{+} ∪ {0}). By the same argument of (4.12) we also obtain

E(ρ_{2}, B(x, r_{2})) ≤ Cε^{\frac{1}{p} ( \frac{2}{p} +p)}.

Then, similar to the derivation of (4.1) we can see that

E_{ε}(u_{ε}, B(x, r_{2})) ≤ \frac{1}{p} ∫_{B} |∇w|^{p}dx + Cε^{\frac{2}{p^{2}} ( \frac{2}{p} +p)}.

At last, by processing as the proof of (5.3) we have

∫_{B(x,r_{2})} |∇h|^{p}dx + ∫_{B(x,r_{2})} |∇u_{3}|^{p}dx + \frac{1}{ε^{p}} ∫_{B(x,r_{2})} u_{3}^{2}dx ≤ C(ε^{1−2/p} + ε^{\frac{2}{p^{2}} ( \frac{2}{p} +p)}).

\underline{Step 2.} Replacing (5.3) by the inequality above, and via the similar argument of Step 1, we also deduce that there exist r_{j} ∈ (2R, r_{j−1}) such that for any j = 1, 2, ⋯,

∫_{B(x,r_{j})} |∇h|^{p}dx+∫_{B(x,r_{j})} |∇u_{3}|^{p}dx+ \frac{1}{ε^{p}} ∫_{B(x,r_{j})} u_{3}^{2}dx ≤ C(ε^{1−2/p} +ε^{a_{j}}), (5.4)

where a_{1} = \frac{2}{p} and a_{j} = \frac{2}{p^{2}} (a_{j−1} + p) for j = 2, 3, · · ·. Obviously, \{a_{j}\} is a increasing and bounded sequence. So we see easily that its limit is \frac{2p}{p^{2}−2}. Letting j → ∞ in (5.4) we have proved Theorem 5.1.

Combining Theorem 5.1 and (5.3) yields that Theorem 1.3 is proved.

Acknowledgements. The research was supported by NSF (19271086) and Tianyuan Fund of Mathematics (A0324628)(China).

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(Received February 27, 2004)