ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu
A NOTE ON p(x)-HARMONIC MAPS
BEI WANG, YUZE CAI
Abstract. This article is concerned withLp(x)estimates of the gradient of p(x)-harmonic maps. It is known thatp(x)-harmonic maps are the weak so- lutions of a system with natural growth conditions, but it is difficult to use the classical elliptic techniques to find gradient estimates. In this article, we use the monotone inequality to show that the minimum p(x)-energy can be expressed by theLp(x) norm of a gradient of a function Φ, which is a weak solution of a single equation.
1. Introduction
LetB ={x∈R2:|x|<1}, S1={x∈R2:|x|= 1}. Assume g(x) =xonS1, and p(x)>1 is a smooth function on B. We are concerned with the gradient of p(x)-harmonic maps. A function uis called a p(x)-harmonic map, if it is a weak solution of
−div(|∇u|p(x)−2∇u) =u|∇u|p(x). (1.1) A functionup is called ap(x)-energy minimizer if it is a solution of
inf Z
B
1
p(x)|∇u|p(x)dx:u∈Wg1,p(x)(B, S1) , (1.2) where p(x)∈(1,2). This minimum is also called the p(x)-energy minimum. It is not difficult to see that thep(x)-energy minimizer is ap(x)-harmonic map.
Partial regularity ofp(x)-harmonic maps in the spaceW1,p(x)(B,R2) was given in [4]. When a p(x)-harmonic map u belongs to W1,p(x)(B, S1) with S1-valued boundary datag, it is more complicated to locate the singularities ofu(cf. [9]).
When p(x) ≥2, the class of function Wg1,p(x)(B, S1) is empty (cf. Page xi in [3]), and problem (1.2) does not make sense. There are two penalized methods to investigate thep(x)-energy minimum, which are helpful to understand the local properties ofp(x)-harmonic maps. First, the Ginzburg-Landau type functional
Eε(u) = Z
B
1
p(x)|∇u|p(x)dx+ Z
B
1
εp(x)(1− |u|2)2dx
can be applied to study p(x)-harmonic maps. Since Wg1,p(x)(B,R2) is not empty, there exists a Ginzburg-Landau minimizer uε. The singularities of p(x)-harmonic
2000Mathematics Subject Classification. 35J56, 35J70, 49J20, 58G18.
Key words and phrases. Gradient estimate;p(x)-harmonic map; drill holes;
minimump(x)-energy.
c
2013 Texas State University - San Marcos.
Submitted April 10, 2013. Published November 29, 2013.
1
maps are often viewed as the limit of zeros ofuε(cf. [9]). Some papers studied thep- energy minimum by estimatingEε(uε) whenpis a constant (cf. [1], [2, 7, 8, 10, 13]).
Second, we can use the method of drilling holes which was introduced in [3, Page xii] to deal with the case ofp(x) non-constant. For example, we can consider the problem
inf Z
Bρ
1
p(x)|∇u|p(x)dx:u∈Wg1,p(x)(Bρ, S1);u|∂B(0,ρ)= x
|x| (1.3) instead of the problem (1.2), sinceWg1,p(x)(Bρ, S1)6=∅. HereBρ=B\B(0, ρ).
In this paper, we investigate the p(x)-energy minimum with p(x) > 2 by the second penalized method. This research is motivated from two aspects. On the one hand, the energy functionalR
Bρ|∇u|p(x)dxcan be used in the theories of phase transitions, such as the problems of superconductivity and superfluids. In the study of type-II superconductors, the vortices can be described by this hole B(0, ρ) (cf.
[3]). On the other hand, the energy functionalR
Bρ
1
p(x)|∇u|p(x)dxcan be used in the partial regularity ofp(x)-harmonic maps. In general, thep(x)-energy minimizer is a p(x)-harmonic map on the domainBρ. The singularities ofp(x)-harmonic maps are also located in those holes.
Since (1.1) is a system with the natural growth condition, it seems difficult to estimate the weak solution by the classical elliptic technique. Now, we show that the p(x)-energy minimum can be expressed by theLp(x) norm of a gradient of a function Φ, which is a weak solution of a single equation. Then, the complicated partial regularity of p(x)-harmonic maps can be understood well by investigating the regularity of a weak solution of such a single equation.
Another problem is whetherx/|x|is ap(x)-harmonic map. In general, the solu- tion of (1.3) exists and is also ap(x)-harmonic map whenp(x) is constant. However, a calculation shows the interesting result: ifp(x) = ˜p(r, θ) depends onθ, thenx/|x|
is notp(x)-harmonic, and hence it does not minimize thep(x)-energy.
2. Main results and proofs
When p(x) is constant and is in (1, n), papers [5, 6, 11] show that x/|x| is a p-energy minimizer, and hence is ap-harmonic map. The following result shows that ifp(x) is variable, thenx/|x|may be not ap(x)-harmonic map.
Theorem 2.1. Let p(x) >1 be a C1(B) function. Assume p(x) = ˜p(r, θ), then x/|x|is ap(x)-harmonic map onB\ {0} if and only ifp(r, θ)˜ is independent ofθ.
Namely, it is a C1 function with one variabler∈[0,1].
Proof. In polar coordinates, u(x) = x/|x| = (cosθ,sinθ). In this case, the p(x)- harmonic maps equation
−div(|∇u|p(x)−2∇u) =u|∇u|p(x) is equivalent to
−div(|∇θ|p(x)−2∇θ) = 0.
Noting ∆θ= 0, the equality above is true if and only if
−∇(|∇θ|p(x)−2)· ∇θ= 0.
Namely,
r2−p(x)logr(∇p(x)∇θ) + (p(x)−2)r1−p(x)(∇r∇θ) = 0.
In view of ∇r∇θ = 0, the result above is equivalent to ∇p(x)∇θ = 0. In polar coordinates,∇θ= (0,1r), then∇p(x)∇θ= 0 is equivalent to
∂θp(r, θ) = 0,˜
which holds if and only if ˜pis independent ofθ. The rest of the proof is not difficult
to complete.
Hereafter, we assumep(x) is independent ofθ. We consider a more general class of functions than those in (1.3),
V1={v∈W1,p(x)(Bρ, S1) : deg(v, ∂B) = 1,deg(v, ∂B(0, ρ)) = 1}.
The main result in this paper, stated below, shows that thep(x)-energy minimum can be expressed by theLp(x)norm of the gradient of
Φ(x) := arctanx2 x1
.
Theorem 2.2.
minnZ
Bρ
1
p(x)|∇v|p(x)dx, v∈V1
o
= Z
Bρ
1
p(x)|∇Φ|p(x)dx. (2.1) Proof. Step 1. We claim Φ(x) = arctanxx2
1 solves the equation
−div(|∇φ|p(x)−2∇φ) = 0, in Bρ; (2.2) and there holds
∇Φ(x)·τ= 1
|x|. (2.3)
In fact, by a simple calculation, we have∇Φ(x) = (−x2, x1)/|x|2. Therefore, (2.3) is true, and
−div(|∇Φ|p(x)−2∇Φ) =−div[(−x2, x1)
|x|p(x) ]
= (x2,−x1)·[log|x|
|x|p(x)∇p(x) +p(x) x
|x|p(x)+2].
Sincep(x) depends only on|x|, we have (x2,−x1)· ∇p(x) = 0. Thus, Φ solves (2.2).
Step 2. Letv∈V1. Set
D= (−v∧vx2+ Φx2, v∧vx1−Φx1), then divD= 0.On the other hand,
D·ν =−(v∧vτ) + Φτ,
whereν is a unit outward norm vector on the corresponding boundary, andτ is a unit tangent vector on the corresponding boundary. Noting (2.3), we have
Z
∂B(0,ρ)
Φτds= 2π.
In view of the definition of the degree d= 1
2π Z
∂B(0,ρ)
v∧vτds= 1,
we obtain
Z
∂B(0,ρ)
(D·ν)ds= 0, Therefore, according to [3, Lemma I.1], there exists
H ∈ {H;D= (Hx2,−Hx1)}.
Step 3. Set
V3(v) =W01,p(x)(Bρ)∩ {H;D= (Hx2,−Hx1)}.
ThenV3(v)6=∅in veiw of 0∈V3(v).
IfV3(v) ={0}, then for anyv∈V1,|∇v|2=|∇Φ|2. Thus, (2.1) holds.
IfV3(v)\ {0} 6=∅, then we can find
H ∈V3(v)\ {0} (2.4)
such that
v∧vx1 = Φx1−Hx1; v∧vx2 = Φx2−Hx2. This means
|∇v|p(x)=|∇(Φ−H)|p(x). Step 4. We claim that
|∇v|p(x)≥ |∇Φ|p(x)−p(x)|∇Φ|p(x)−2∇Φ· ∇H. (2.5) To prove this inequality, we define the function
f(s, t) =|t−s|p(x)− |t|p(x)+p(x)|t|p(x)−2(t·s)
for two vectorssandt. According to the mean value theorem, there existsξ∈(0,1) such that
|t|p(x)− |t−s|p(x)=p(x)|t−ξs|p(x)−2(t−ξs)·s.
Hence, applying the monotone inequality [14, (2.11)]), we have f(s, t) =p(x)|t|p(x)−2(t·s)−p(x)|t−ξs|p(x)−2(t−ξs)·s
=p(x)ξ−1[|t|p(x)−2t− |t−ξs|p(x)−2(t−ξs)]·[t−(t−ξs)]
≥γ0|s|p(x)≥0.
Here γ0 >0 only depends on p(x). Taking s=∇H, t =∇Φ, and by Step 3, we can see (2.5).
Step 5. For anyv∈V1, (2.5) implies that Z
Bρ
1
p(x)|∇v|p(x)dx≥ Z
Bρ
1
p(x)|∇Φ|p(x)dx− Z
Bρ
|∇Φ|p(x)−2∇Φ· ∇Hdx. (2.6) Since H ∈W01,p(x)(Bρ), and Φ is a solution of (2.2), we see that the second term of the right-hand side of (2.6) is zero. Hence, (2.6) leads to
Z
Bρ
1
p(x)|∇v|p(x)dx≥ Z
Bρ
1
p(x)|∇Φ|p(x)dx, which implies
inf
V1
Z
Bρ
1
p(x)|∇v|p(x)dx≥ Z
Bρ
1
p(x)|∇Φ|p(x)dx. (2.7)
Step 6. Letu∗= (cos Φ,sin Φ) with Φ(x) = arctanxx2
1. Then, (2.3) impliesu∗∈V1. Clearly,
Z
Bρ
1
p(x)|∇u∗|p(x)dx= Z
Bρ
1
p(x)|∇Φ|p(x)dx.
Thep(x)-energy minimum attains R
Bρ|∇Φ|p(x)dx at this functionu∗. Combining
with (2.7), we complete the proof.
Acknowledgments. This work was supported by Natural Science Foundation of Jiangsu Higher Education Institutions (No.13KJB110003), and by the 12th five- year plan of Jiangsu Institute of Education (Jsie2012yb02).
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Bei Wang
School of mathematics and information technology, Jiangsu Institute of Education, Nanjing, Jiangsu 210013, China
E-mail address:[email protected]
Yuze Cai
Department of Basic Science, Shazhou Professional Institute of Technology, Zhangji- agang, Jiangsu 215600, China
E-mail address:[email protected]
