Regularity of the m-symphonic map

We introduce a new energy functional of conformal invariance and consider its critical points, named the m-symphonic map. We study a Hölder continuity of m-symphonic maps from domains of Rm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^m$$\end{document} into the spheres in the higher dimension m≥4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \ge 4$$\end{document}.


Introduction
Symphonic maps are critical points of the energy functional, called symphonic energy, is a measurement of the conformality of a map from (M, g) to (N, h), since T u ¼ 0, if a map u is conformal, where m is the dimension of the source manifold M. The maps u satisfying T u ¼ 0 is called weakly conformal, that is, du ¼ 0 or u is conformal. The squared integral of T u is introduced to look for maps closest to conformal ones. The first and second variation formulas, monotonicity type formula and Bochner type formula are derived and critical points of the functional are geometrically studied, relating to the conformality and homotopy of maps [15,20,21]. T u is also equal to the stress energy tensor in the two dimension case m ¼ 2 (refer to [3, 4, p. 392]). Both of the 4-energy and the symphonic energy (1.1) are conformally invariant in the four dimensional domain. While the symphonic energy density is the norm of the pullback u Ã h, the 4-energy density is the trace of the pullback u Ã h. Thus, the symphonic energy may have more geometric information than that of the 4-energy. That is our motivation of studying the symphonic maps. The higher dimensional m ! 4 analogue to (1.1) is the following functional on the mdimensional domain, called m-symphonic energy, which is indeed invariant under any conformal changes of metrics of the m-dimensional domain. Critical points of m-symphonic energy is called m-symphonic maps. The msymphonic maps naturally concern the usual m-harmonic maps, as explained above, and also the higher dimensional hypersurface of prescribed mean curvature, that reflects the regularity proof by use of the determinant structure of nonlinear term (see [14,23,26]). In this paper we prove the following regularity for m-symphonic maps, m ! 4, that extends the previous result for symphonic maps in [18] to the higher dimension m ! 4: Main Theorem (Theorem 5.1) Let m ! 4 and X be a domain of the m-dimensional Euclidean space R m . Then any m-symphonic map from X into the standard sphere is Hölder continuous.
The main theorem follows from a similar argument as in [18], of that the key ingredient is the determinant structure of the nonlinear term coming from the target constraint. The compensatory regularity of the determinant is nowadays well known and is applied to geometric equations such as harmonic maps, originally, two or higher dimensional surface of prescribed mean curvature. See [5,12] for harmonic maps, [25,27] for p-harmonic maps, [11] for polyharmonic maps, [22] for a new Sobolev type inequality and [16] for the H-system heat flow. However, here an algebraic property specific to the p-symphonic operator is emphasized, that is the following monotonicity inequality (Proposition 3.1): . . . ; A n Þ and B ¼ ðB 1 ; B 2 ; . . . ; B n Þ, where rð Þ denotes the symphonic energy density form: The coercive inequality above for the p-symphonic operator is clearly different from that of the p-harmonic operator. The m-symphonic energy (1.2) is an m/2 powered integral of the norm of the pullback u Ã h, on the other hand, the well-known m-harmonic maps are critical points of the m-energy, an m/2 powered integral of the trace of the pullback u Ã h, The coerciveness of the p-symphonic operator seems to stem from the norm of pullback metric. Refer to the proof of Proposition 3.1. The proof of regularity as in the main theorem follows from the local energy growth estimate, Theorem 5.2 below. The msymphonic maps on the m-dimensional domain is on the energy critical case in the sense that the local m-energy on a m-dimensional ball is invariant under the usual scaling transformation (see Theorem 5.2, 5.1, 5.6) in Sect. 5). Thus, the regularity criterion may naturally be given by the local energy, similarly as the m-harmonic maps on the mdimensional domain. Such regularity argument as above have already been performed in the case that m ¼ 4 by [17,18]. Thus, the process and ingredients of the proof in the higher dimension m ! 4 here are similar to that of [18]. Based on the usual rescaling argument, the classical regularity tools well-known nowadays are employed, such as Hardy and BMO estimates with the duality between the Hardy space and BMO space (see Sect. 4). Such regularity methods have been applied for harmonic and p-harmonic maps (refer to [5,12,25,27]). Here, we shall show how to make use of the algebraic structure of the principal part in the p-symphonic maps, presented above, in the regularity proof.

Preliminaries
We gather the notations, used in the followings. Let X be a domain in R m , m ! 1, and p [ 1. The Sobolev space of R n -valued L p -functions on X is denoted by L 1; p X; R n ð Þ, n ! 1, where dx is the Lebesgue measure, and D i u a , i ¼ 1; . . .; m; a ¼ 1; . . .; n, are weak derivatives and DuðxÞ ¼ D i u a ðxÞ ð Þis the gradient. L 1; p 0 ðX; R n Þ is the completion on L 1; pnorm of R n -valued smooth maps with compact support on X. The S nÀ1 -valued Sobolev maps on X, L 1; p ðX; S nÀ1 Þ, is defined as be The p-symphonic energy is introduced as follows: Let p > 1. The p-symphonic energy is defined as is the symphonic energy density.

Remark 2.1
The p-symphonic energy density is of p-th powered gradients. In fact, there exists a positive constant C [ 1 such that 1 C jDuj p ju Ã hj p=2 CjDuj p : ð2:1Þ A weak solution of p-symphonic maps from X into S nÀ1 is given as follows: Now we consider a weak solution of The operator in left hand side of (2.4) is called as p-symphonic operator. We shall present the Hölder regularity of a weak solution of (2.4), for later use. Proof We argue as in the proof of Theorem 1 in [17] (also refer to [10,19]). The growth estimate of local energy in (2.5) is shown to hold for a local minimizer. If the target manifold ðR n ; hÞ has the Euclidean metric h ¼ d ab À Á , then the Euler-Lagrange equation is (5.14) and a weak solution of (5.14) is a local minimizer. Putting and thus, its quadratic form is where the Cauchy inequality is used for the second term of the right hand side in the first equality. So, the local minimality of a weak solution follows from the Taylor expansion. The Hölder regularity follows from Morrey's Dirichlet growth theorem with (2.5), as usual (see [9,10,19]). h

A monotonicity inequality for p-symphonic operator
Here, an algebraic inequality associated with the p-symphonic maps is presented, that will be crucially used in the regularity proof later.
. . .; m, let rð Þ denote the symphonic energy density form : and that ð3:3Þ Proof Put the notation Noting that a ij and b ij are positive semi-definite, we have Replacing A by B we get which is furthermore estimated below by (3.4, 3.5) The above quantities are arranged and computed as The last inequality follows from the non-negativity of the second term in the previous SN Partial Differential Equations and Applications quantity, since p ! 4. The inequality (3.2) is obtained. For the proof of (3.3), put the notation By direct computation we have XðtÞ kl a kl À b kl ð Þdt and thus, XðtÞ kl a kl À b kl ð Þdt XðtÞ kl a kl À b kl ð Þ where, in the last inequality, we use the Schwarz inequality in R m 2 to estimate X m k;l¼1 XðtÞ kl a kl À b kl ð Þdt The proof of (3.3) is complete. h

Function spaces
The Hardy space and the space of functions of bounded mean oscillation are crucially used in the proof of the main theorem. Here let us recall the definitions and properties of these function spaces (See more details in [7,24,28]).    where the positive constant C depending only on n.

SN Partial Differential Equations and Applications
Finally, we require the Hardy space estimate in L p -setting, due to Coifman, Lions, Meyer and Semmes (refer the proof in [2,5]). Lemma 4.2 Suppose that u 2 L 1; m ðR n ; RÞ and v ¼ v i ð Þ 2 L m=ðmÀ1Þ ðR n ; R n Þ, and furthermore,

Hö lder continuity of m-symphonic maps
In this section we shall prove our main theorem. Proof of Theorem 5.2 We proceed the proof by reduction to absurdity. Suppose that, for any positive number h\1, there exist families of weak solutions fu k g and balls Bðx k ; r k Þ & X, k ¼ 1; 2; . . ., such that ð5:9Þ By (5.8), the sequence fv k g is bounded in L 1; m ðBð1Þ; R n Þ and thus, there exist a subsequence fv k g, denoted by the same notation as above, and the limit map v 1 2 L 1; m ðBð1Þ; R n Þ such that for any smooth map / defined on B(1) into R n , with compact support. By (5.6) it is clear that almost every x 2 Bð1Þ: ð5:13Þ Now we assert that the limit map v 1 satisfies Z for any R n -valued smooth map / on B(1) with compact support. The proof will be given in Sect. 6 Here we shall claim the strong convergence of gradients, that is proved in Sect. 7, jDv k j 2 À jDv 1 j 2 2 ! 0 strongly in L 1 ðBð1=2Þ; RÞ: Then, taking the limit k ! 1 in (5.9) forces, for h\1=2, lim k!1 Z BðhÞ jDv 1 j mÀ4 jDv 1 j 4 dx ! 2 À1 ; contradicting to the limit taken in (5.15) as k ! 1. h

Weak compactness
In this section we show the validity of (5.14), that is, the limit map v 1 obtained in (5.10) and (5.11) verifies (5.14).
Then it holds that Z Bð1Þ rðDvÞ ðmÀ4Þ=4 X m i; j¼1 for any R n -valued smooth map / with compact support in B(1).
The proof relies on a weak compactness, that is the convergence in the distributional sense of our m-symphonic operator under that of sequence fv k g in (5.11) and (5.10) (refer to [6] for harmonic case and [1,8,30] for p-Laplacian case).

ð6:3Þ
Proof For the symphonic maps with p ¼ 4 the similar assertion as above is obtained in [18], following the argument in [1, Theorem 2.1, pp. 31-33]. Here, the proof is almost same as that in in the symphonic maps case p ¼ 4, except for the use of algebraic inequalities. Thus, we indicate only how to use the algebraic inequalities Proposition 3.1 peculiar to the p-symphonic operator. Let, for any positive d\1 and k ! 1, The estimation in [18, (5.7), Page 12] is changed in the following: By use of the algebraic inequality (3.2) in Proposition 3.1 we compute as ð6:5Þ The estimation of each term in the right hand side of (6.5) is performed similarly as in [18,. h Now the proof of Theorem 6.1 follows from Lemma 6.1.
Proof of Theorem 6.1 Let / be a R n -valued smooth map with compact support in B(1). We compute as The second term in the right hand side of (6.6) simply converges to zero as k ! 1, by the weak convergence (5.11), since, with m 0 ¼ m mÀ1 , By virtue of (6.3) in Lemma 6.1, the first term in the right hand side of (6.6) is estimated above as Here, the first inequality of (6.7) is by the Schwarz inequality for Euclidean inner product on R m 2 . The second inequality follows from the algebraic inequality (3.3) in Proposition 3.1. In the last inequality of (6.7), the Hölder inequality is used with exponent l as m m À 1 l\2 ¼) l l À 1 m and thus, the second term in the right hand side of (6.7) is bounded by (5.8) and, by (6.3) in Lemma 6.1, the first term goes to zero as k ! 1.

Strong compactness
In this section we will show the strong convergence of fDv k g in (6.3), Here the maps v k and v 1 are extended into all of the space R m by letting v k ¼ 0 and v 1 ¼ 0 outside Bð1Þ ¼ Bð0; 1Þ. Let g be a real-valued smooth function on R m such that 0 g 1, supp g & Bð5=8Þ and g ¼ 1 in B(1/2). In the following we often the abbreviate as B(r) a ball Bð0; rÞ with center of the origin 0 and radius r [ 0. Now we present the BMO estimate, that is similar to [18, Lemma 6.1, Page 16] (also see [5]).
Lemma 7.1 The sequence fg v k g is bounded in BMOðR m ; R n Þ.
Now we define the vector field on B(1) as À! 0 strongly in L 1 ðBð1=2Þ; RÞ: ð7:3Þ Proof Subtracting (5.14) for the limit ð7:6Þ where we use by (5.6) and (5.13) that Dw k ¼ Eðx k ; r k Þ 1=m Dv k ; D i v k Á w k ¼ Eðx k ; r k Þ À1=m X n a¼1 D i w a k w a k ¼ Eðx k ; r k Þ À1=m 2 À1 D i w k j j 2 ¼ 0:

ð7:7Þ
By the Fefferman-Stein inequality, Lemma 4.1, the first term in the right hand side of (7.6) is bounded for each a; b ¼ 1; . . .; n by Z ð7:8Þ where Lemmas 7.1 and 7.3 are crucially used and, g v is in BMOðR m ; R n Þ, by the Poimcaré inequality, (5.8) and (5.11). The second term in the right hand side of (7.6) is estimated above for each a; b ¼ 1; . . .; n as Z