Continuity of Almost Harmonic Maps with the Perturbation Term in a Critical Space

Mati ur RAHMAN Yingshu L Deliang XU

Abstract The authors study the continuity estimate of the solutions of almost harmonic maps with the perturbation term f in a critical integrability class (Zygmund class)L, n is the dimension with n ≥3. They prove that when q > the solution must be continuous and they can get continuity modulus estimates. As a byproduct of their method, they also study boundary continuity for the almost harmonic maps in high dimension.

Keywords Harmonic maps, Nonlinear elliptic PDE, Boundary regularity

1 Introduction

Let B ?Rnbe an open ball and(N,h)be a smooth Riemannian manifold which is compact and without boundary. We may assume that N is isometrically embedded into the Euclidean space Rmby the Nash’s embedding theorem. Consider the Dirichlet functional

Its critical points are called harmonic maps and satisfy the Euler-Lagrange equation

where A is the trace of second fundamental form of (N,h).

The study of regularity for harmonic maps has a long history which can be traced to Morrey[12] for two dimension case and to Schoen and Unlenbeck [22] for higher dimensions. In the two dimensional case, because of the conformal invariance property, the analysis for regularity of weak solutions of harmonic maps was pioneered by Hein [5–6] who proved that every weakly harmonic map from a surface into a compact manifold is always smooth. Later, these results were extended to higher dimensions by Evans [3] for the target manifold which is a sphere,and Bethuel[1]for the general case,they proved partial regularity results for stationary harmonic maps by using similar ideas of H′elein. Recently, Rivi`ere [19] found a new approach to study the regularity of the solution of conformally invariant two dimensional geometric variational problems, which include harmonic maps from two dimensional domain and the famous Hildebrandt’s conjectures. In this new approach (see [19]), a key observation is that system (1.1) can also be written as the following more general form

where ? is an antisymmetric matrix and(1.2)is called as Rivi`ere’s equation,please see[16]and[24] for more details. In a similar way, Rivi`ere and Struwe [20] extended this method to high dimensional regularity of harmonic maps under the assumption of smallness of the solution in some homogeneous Morrey space. This method also has some other applications(see[8,16,26]).Another kind of elliptic systems sharing the structure like (1.2) are so called Dirac-harmonic map, which is inspired by the supersymmetric nonlinear sigma model from the quantum field theory, and is a natural and interesting extension of harmonic maps in an analytic literature.Related studies for regularity of Dirac-harmonic map are referred to [2, 29].

Almost harmonic maps (Approximation of harmonic map), mean harmonic maps with a perturbation (or a potential) term f:

in B, a bounded domain of Rn. Here f : B →Rmis a vector function in some suitable Euclidean space Rm. Actually, to compare with (1.2), we can study more general elliptic systems as Rivi`ere’s equation by adding a potential term f:

where ? is an antisymmetric one form valued matrix and belongs to L2.

The study of almost harmonic maps, to our knowledge, comes from two aspects.

On one hand,from the definition of the harmonic maps,it is natural to find critical points of the Dirichlet energy. However,the classical variational methods cannot be used to the Dirichlet energy because E(u) does not satisfy the Palais-Smale condition. Sacks and Uhlenbeck [21],Lamm[7] introduced a regularization of the Dirichlet energy to overcome this difficulty. Later,Lin and Wang[10–11]used a Ginzburg-Landau approximation to regularize the Dirichlet energy and proved the energy monotonicity formula in this case. In this paper,we consider the equation in a bounded domain B,

The energy functional F(u) of this Euler-Lagrange equation is

for some V ∈C1(N), in this case, f =?V(u).

On the other hand, it is well-known that for the functional E(u), the harmonic map heat flow is the L2gradient flow. The corresponding equation is

Due to its restriction of weak solution, we consider some class of weak solutions which satisfies an energy identity. Then it holds

If some solution of (1.5) satisfies this inequality and the initial data has finite energy, we have that almost any time slice satisfies

From this result, it is of our interest to study the almost harmonic maps with perturbation in the different spaces. Moser[14] considered the perturbation term f ∈Lp, p>and proved Hlder continuity for weak solutions under a suitable smallness condition. Similarly, for the same case, Sharp and Topping [24] used a type of “geometric bootstrapping” and iteration method which can show that the solutions have regularity property in two dimension. Also in high dimensions,Sharp[23]used the coulomb gauge to show the improved regularity. For p=Moser [15] obtained an inequality in an Orlicz space belonging to a function with exponential growth. Later,the regularity results were extended to higher dimensions with p ∈(1,∞),under an appropriate smallness condition, a certain degree of regularity follows in [16]. Li and Zhu[9] considered f ∈L ln+L and proved the compactness of mapping from Riemannian surface with tension fields which are bounded in L ln+L. Later, Sharp and Topping [24] extended the results of Li and Zhu and showed the stronger compactness results under the condition of f merely bounded in L ln L.

For the almost harmonic maps in high dimensions, the proof holds always with the help of a suitable smallness conditions. We know that the well-known monotonicity formula (see[18]) can be applied to prove the stationary condition changing to smallness of the energy of solutions, this way would not have an influence on the expected results. In general, there is no monotonicity formula for the almost harmonic maps, however, Struwe [25] found that monotonicity formula can be viewed as a parabolic version for the harmonic map heat flow.

In this paper,we consider the regularity properties for the weak solutions of almost harmonic maps with perturbation in a critical Zygmund class,or specific Orlicz space LplogqL. We show that we have this type of regularity which is similar to the regularity results of harmonic maps under suitable smallness conditions.

Our main results are as follows.

Remark 1.1Here the exponentis critical in the sense that even for linear equation,?u = f, we cannot expect the continuity of the solution. In this critical exponent level, we consider continuity problem by assuming that f belongs to a Zygmund spacelogqL. Indeed,we prove the related regularity result for system (1.4), see Theorems 2.1 and 3.1.

Throughout this paper, we use the convention of the summation. The standard Lebesgue spaces are denoted by Lp(B)(p ≥1 and B is a domain of Rn). Br(x)denotes the ball of radius r > 0 around the center x ∈Rnand |Br(x)| denotes Lebesgue measure (volume). The mean value of some function f(x) over Br(x) is defined as

Various constants arise in our paper unless indicated otherwise, they are always absolute constants. The symbol C denotes a generic constant and its value may change from line to line.

2 Interior Regularity and Proof of Theorem 1.1

At the analytical level, our motives, to derive the regularity property from the log part integrability factor q, come from the following improved Morrey lemma.

Lemma 2.1Suppose that p ≥1 and α>1. There exists a constant C0, depending only on n,α and A, such that the following holds. Suppose u ∈W1,p(B2R(x0)) satisfies

for every x1∈BR(x0) and 0

ProofFirst noting that the Hlder inequality and (2.1) imply that

Now letting rk=2?kr0, similarly we have

and similar estimates hold for y2instead of y1. By Lebesgue’s differential theorem, we know that for almost every y1(and similarly for almost every y2)

Then by summing up the above inequalities, we obtain

This implies (2.2) and completes the proof.

Let p ≥1 and q ∈R. We define the Orlicz norm as

and

Lemma 2.2Let f ∈logqL(BR), n > 2, q ≥0 and ? ∈L∞(BR). Then there exists R0>0 such that when 0

ProofBy the well-known Hlder inequality for LplogqL space (or the duality of Orlicz space), we have

here χ(BR) denotes the characteristics function of BR. Now we look for the solution of the equation

however it is easy to see that

so by using the intermediate value theorem, we conclude (2.10) is true. Then by (2.8) and the definition of Orlicz norm for χ(BR), we have

this completes the proof.

Let B be a bounded domain in Rn, recall that a function f ∈(Rn)belongs to the space BMO(B) if

We need the following lemma by Unlenbeck [28] or Rivi`ere and Struwe [20] (optimal gauge transformation).

Lemma 2.3There exist ε(n) > 0 and C(n) such that, for every ? = (?αβ)1≤α,β≤min L2(B1,so(n)?Rm) satisfying

there exist ξ ∈W1,2(B1,so(n)?Λn?2Rn) and P ∈W1,2(B1,SO(n)) such that

1)

2)

3)

First we notice that the harmonic map equation(1.3)can be rewritten as the elliptic system(1.4), the details of these deductions we refer to [20] for codimension one case and [16] for general case. So we just focus on proving the regularity of elliptic system (1.4).

We have the following result.

then u is continuous in the interior of B.

ProofThe proof will be divided into several steps. Since the regularity is a local property,we assume for simplicity that (B,g) = (B1,g0), where B1?Rnis the unit ball with the standard Euclidean metric g0in Rn. Let u be a weak solution of (1.4),

Then using integration by parts and the duality of Hardy space H and BMO space, we have

then from (2.14),

By using the integration by parts and combining with (2.12), (2.15) and (2.16),

and Lemma 2.2 implies that

Finally we establish the estimate for the harmonic 1-form h in a standard way by using the Campanato estimates result for harmonic functions of Giaguinta,see[4,Theorem 2.1 on p. 78],which yields for any 0

Then combining (2.18), (2.20) and (2.21) together, we obtain

Multiplying by rp?nand, for brevity, denoting by

and

then (2.23) implies that

On the other hand, we can estimate the term ‖u‖BMO(BR(z))by a well-known fact, see [20],

Hence we obtain

Now, pick some 0<θ0

Choosing θ0small enough to ensure C1and then choosing ε0small enough such that ε0≤, we get the estimate

Similarly letting rk=2?kr0, we have

and similar estimates hold for x2instead of x1. By Lebesgue’s differential theorem we know that for almost every x1(and similarly for almost every x2)

Then summing up above inequalities together and using (2.30) again, we obtain

(2.33) is satisfied for all x1x2∈B1, 4r0< R1, andBecause1, (2.33)implies that u is continuous onand this completes the proof.

3 Continuity Estimate up to the Boundary and Proof of Theorem 1.2

In this section,we establish the regularity of the solution of(1.3)or(1.4)up to the boundary.To derive the continuity of u up to boundary we first need to get the following variant Dirichlet type growth theorem, which gives an appropriate estimate for the modulus of continuity for u.

Remark 3.1Indeed, we can prove this result by using similar argument as in the proof of (2.30) by combining with the argument for the proof of Lemma 2.1, however, here we use a technique from Morrey in [13], which is also used by M¨uller and Schikorra [17] for proving the boundary regularity result for similar problem in the two-dimensional case.

ProofFor any z ∈Br0(x0), we have

hence

Denoting by xt=x+t(x0?x) for t ∈[0,1], a direct calculation and using (2.30) we obtain

here we choose 0<γ <1, so r0t ≤2r0tγfor all t ∈[0,1], which implies (3.1).

Now we are in a position to give the proof of Theorem 1.2. Similarly as for interior regularity,we prove the following theorem for system (1.4), then Theorem 1.2 can be deduced as an application.

then u is continuous up to the boundary of B1.

By assumption of theorem, representation of the trace u|?B1= φ(Θ) is continuous. Let us fix x0= Θ0∈?B1and let x1= (r1,Θ1) be an interior point in B1. Let x?= (r1,Θ?) ∈Bδ2(x1),where Θ?will be chosen later and δ =1 ?r1. Denoting by

then we have

It is easy to see that for small enough δ and small|Θ0?Θ1|, the term III becomes small. From Proposition 3.1 and (2.30), we have

so we have

this contradicts with (3.2). Hence we have

and this completes the proof.

Remark 3.2When dimension is two,similar problems for the study of(1.3)and(1.4)were investigated widely, for example see [9, 17, 23], however we cannot use our results directly to the case dim=2, this is due to that in Lemma 2.2 we need the condition>1. We study the regularity up to boundary and its global compactness properties of (1.4) in two dimensional case in a forthcoming paper.

AcknowledgementThe authors would like to thank Prof. Congming Li for his encouragements and valuable suggestion.