Neck analysis of extrinsic polyharmonic maps

Abstract

We prove the energy identity and the no neck property for a sequence of smooth extrinsic polyharmonic maps with bounded total energy.

Appendices

Appendix 1: $\varepsilon$-regularity

In this section we prove the $\varepsilon$-regularity theorem for (extrinsic) m-polyharmonic maps.

Theorem 4.2

($\varepsilon$-regularity) Suppose $u\in W^{2m,p}(B,N), p>1$, is an m-polyharmonic map from the unit ball $B\subset \mathbb {R}^{2m}$ to a compact Riemannian manifold N isometrically embedded into $\mathbb {R}^K$. There exists $\varepsilon _0>0$ such that if

$$\begin{aligned} E(u;B)\mathpunct {:}=\frac{1}{2}\int _{B}\left( |\nabla ^mu|^2+|\nabla u|^{2m}\right) \le \varepsilon _0, \end{aligned}$$

then for some constant C,

$$\begin{aligned} \Vert u-{\bar{u}}\Vert _{W^{2m,p}(B_{1/2})}&\le C\left( \Vert \nabla ^mu\Vert _{L^2(B)}+\Vert \nabla u\Vert _{L^{2m}(B)}\right) , \end{aligned}$$

(4.12)

and for $l=1,2,\ldots$,

$$\begin{aligned} \Vert \nabla ^lu\Vert _{L^\infty (B_{1/2})}&\le C(l)\left( \Vert \nabla ^mu\Vert _{L^2(B)}+\Vert \nabla u\Vert _{L^{2m}(B)}\right) , \end{aligned}$$

(4.13)

where ${\bar{u}}$ is the integral mean of u over $B\subset \mathbb {R}^{2m}$ and $C, \left\{ C(l)\right\}$ are constants.

Remark

We will show that the Sobolev norm $\Vert u\Vert _{W^{m,2}(B)}$ can be controlled by E(u; B) provided that ${\bar{u}}=0$. In fact, since $u\in W^{2m,p}(B)\hookrightarrow W^{m,2}(B)$ and N is compact, we know that $u-{\bar{u}}\in W^{m,2}(B)$. The interpolation inequality (see [1, Lemma 5.2(1)]) applied to $u-{\bar{u}}$ gives

$$\begin{aligned} \Vert \nabla ^ju\Vert _{L^2(B)}\le C\left( \Vert \nabla ^mu\Vert _{L^2(B)}+\Vert u-{\bar{u}}\Vert _{L^{2}(B)}\right) ,\quad \forall 1<j\le m. \end{aligned}$$

Combining it with Poincaré inequality

$$\begin{aligned} \Vert u-{\bar{u}}\Vert _{L^2(B)}\le C\Vert \nabla u\Vert _{L^2(B)}, \end{aligned}$$

we obtain

$$\begin{aligned} \begin{aligned} \Vert \nabla ^ju\Vert _{L^2(B)}&\le C\left( \Vert \nabla ^mu\Vert _{L^2(B)}+\Vert \nabla u\Vert _{L^2(B)}\right) \\&\le C\left( \Vert \nabla ^mu\Vert _{L^2(B)}+\Vert \nabla u\Vert _{L^{2m}(B)}\right) ,\quad \forall 1<j\le m. \end{aligned} \end{aligned}$$

(4.14)

If in addition, we assume that ${\bar{u}}=0$, then the Poincaré inequality reads

$$\begin{aligned} \Vert u\Vert _{L^2(B)}\le C\Vert \nabla u\Vert _{L^2(B)}, \end{aligned}$$

and

$$\begin{aligned} \Vert u\Vert _{W^{m,2}(B)}^2=\sum _{j=0}^{m}\Vert \nabla ^ju\Vert _{L^2(B)}^2\le C\sum _{j=1}^m\Vert \nabla ^ju\Vert _{L^2(B)}^2. \end{aligned}$$

Since (4.14) holds trivially for $j=1$, we conclude that

$$\begin{aligned} \Vert u\Vert _{W^{m,2}(B)}\le C\left( \Vert \nabla ^mu\Vert _{L^2(B)}+\Vert \nabla u\Vert _{L^{2m}(B)}\right) , \end{aligned}$$

(4.15)

provided that ${\bar{u}}=0$.

Proof

Without loss of generality, we may assume that ${\bar{u}}=0$. Since u is an (extrinsic) m-polyharmonic map, it satisfies the following Euler–Lagrange equation (see [2, Lemma 2.2, (4)])

$$\begin{aligned} \Delta ^mu=\sum _{\begin{array}{c} i,j,q\ge 0\\ i+j+q=m-1\\ (i,j,q)\ne (0,m-1,0) \end{array}}c_{ijq}^{m-1}\Delta ^i\nabla ^q(P(u))\Delta ^{j+1}\nabla ^qu -\Delta ^{m-1}(A(u)(\nabla u,\nabla u)),\qquad \end{aligned}$$

(4.16)

where $c_{ijq}^m$ are positive integers, $P(u)=D\Pi (u)$ is the orthonormal projection defined by the differential of the nearest projection map $\Pi :N_\delta \rightarrow N, N_\delta \subset \mathbb {R}^K$ is a neighborhood of N and A(u) is the second fundamental form of $N\hookrightarrow \mathbb {R}^K$ at u.

For a multi-index $\alpha =(k_1,k_2,\ldots , k_a)$, where $k_i\ge 1$ are integers for $i=1,2,\ldots ,a$, the norm of $\alpha$ is defined to be $|\alpha |\mathpunct {:}=\sum _{i=1}^ak_i$ and a is called the length of $\alpha$. We can rewrite (4.16) as

$$\begin{aligned} \Delta ^mu=\sum _{\mathop {\alpha \ne (2m)}\limits ^{|\alpha |=2m}}a_\alpha (u)\nabla ^{k_1}u\#\nabla ^{k_2}u\#\cdots \#\nabla ^{k_a}u, \end{aligned}$$

(4.17)

where $a_\alpha$ are smooth functions on N. Moreover, we may assume that $k_1\ge k_2\ge \cdots k_a\ge 1$ in (4.17). In particular, $1\le k_1\le 2m-1$ and $k_2\le m$.

Now, for fixed $\sigma \in (1/2,1)$, let $2\sigma '=1+\sigma$ and $\phi \in C_0^\infty (B_{\sigma '})$ be a cutoff function satisfying

$$\begin{aligned} \phi |_{B_\sigma }\equiv 1\quad \text {and} \quad |\nabla ^k\phi |\le 4^k(1-\sigma )^{-k},\qquad k=0,1,2,\ldots . \end{aligned}$$

In order to apply $L^p$-estimates to $\phi u$, we compute the equation of $\phi u$ as follows

$$\begin{aligned}&\Delta ^m(\phi u)-\sum _{i=1}^{2m}\nabla ^i\phi \#\nabla ^{2m-i}u =\phi \Delta ^mu=\sum _{\mathop {\alpha \ne (2m)}\limits ^{|\alpha |=2m}}\phi a_\alpha (u) \nabla ^{k_1}u\#\nabla ^{k_2}u\#\cdots \#\nabla ^{k_a}u \\&\qquad =\sum _{\mathop {\alpha \ne (2m)}\limits ^{|\alpha |=2m}}a_\alpha (u)\left[ \nabla ^{k_1}(\phi u)-\sum _{j=1}^{k_1}\left( {\begin{array}{c}k_1\\ j\end{array}}\right) \nabla ^{j}\phi \nabla ^{k_1-j}u\right] \#\cdots \#\nabla ^{k_a}u\\&\qquad =\sum _{\mathop {\alpha \ne (2m)}\limits ^{|\alpha |=2m}}a_\alpha (u)\nabla ^{k_1}(\phi u)\#\cdots \#\nabla ^{k_a}u+\sum _{\mathop {\alpha \ne (2m)}\limits ^{|\alpha |=2m}}\sum _{j=1}^{k_1}a_\alpha (u)\nabla ^{j}\phi \#\nabla ^{k_1-j}u\#\cdots \#\nabla ^{k_a}u. \end{aligned}$$

Thus,

$$\begin{aligned} \begin{aligned} \Delta ^m(\phi u)&=\sum _{i=1}^{2m}\nabla ^i\phi \#\nabla ^{2m-i}u+\sum _{\mathop {\alpha \ne (2m)}\limits ^{|\alpha |=2m}}a_\alpha (u)\nabla ^{k_1}(\phi u)\#\cdots \#\nabla ^{k_a}u\\&\qquad +\sum _{\mathop {\alpha \ne (2m)}\limits ^{|\alpha |=2m}}\sum _{j=1}^{k_1}a_\alpha (u)\nabla ^{j}\phi \#\nabla ^{k_1-j}u\#\cdots \#\nabla ^{k_a}u. \end{aligned} \end{aligned}$$

(4.18)

Next, for $1<p<\frac{2m}{2m-1}$, we estimate the $L^p(B)$ norm of the terms at the right-hand side of (4.18) one by one. For the first term, by the definition of $\phi$,

$$\begin{aligned} \Vert \nabla ^i\phi \#\nabla ^{2m-i}u\Vert _{L^p(B)}\le C(1-\sigma )^{-i}\Vert \nabla ^{2m-i}u\Vert _{L^p(B_{\sigma '})},\quad 1\le i\le 2m. \end{aligned}$$

For the second term, by generalized Hölder inequality,

$$\begin{aligned} \Vert \nabla ^{k_1}(\phi u)\#\cdots \#\nabla ^{k_a}u\Vert _{L^p(B)}&\le C\Vert \nabla ^{k_1}(\phi u)\Vert _{L^{\frac{2mp}{2m-(2m-k_1)p}}(B)}\prod _{i=2}^a\Vert \nabla ^{k_i}u\Vert _{L^{\frac{2m}{k_i}}(B)}\nonumber \\&\le C\Vert \phi u\Vert _{W^{2m,p}(B)}\prod _{i=2}^a\Vert u\Vert _{W^{m,2}(B)}, \end{aligned}$$

(4.19)

where in the last inequality, we have used the following Sobolev embedding

$$\begin{aligned} W^{2m,p}\hookrightarrow W^{k_1,2mp/(2m-(2m-k_1)p)}\quad \text {and} \quad W^{m,2}\hookrightarrow W^{k,2m/k},\quad 1\le k\le m. \end{aligned}$$

The estimate of the last term is obtained by different methods for different values of $k_1$ and j.

• If $j=k_1$, exploiting the boundedness of u and by Sobolev inequalities (see 4.19), we obtain

  $$\begin{aligned}&\Vert \nabla ^{j}\phi \#\nabla ^{k_1-j}u\#\nabla ^{k_2}u\#\cdots \#\nabla ^{k_a}u\Vert _{L^p(B)}\\&\qquad \le C(1-\sigma )^{-k_1}\Vert u\#\nabla ^{k_2}u\#\cdots \#\nabla ^{k_a}u\Vert _{L^p(B)}\\&\qquad \le C(1-\sigma )^{-k_1}\Vert u\Vert _{W^{m,2}(B)}^{a-1}. \end{aligned}$$

• If $\max \left\{ 1,k_1-j\right\} \le j<k_1(<2m)$, then by the definition of $\phi$,

  $$\begin{aligned}&\Vert \nabla ^{j}\phi \#\nabla ^{k_1-j}u\#\nabla ^{k_2}u\#\cdots \#\nabla ^{k_a}u\Vert _{L^{p}(B)}\\&\qquad \le C(1-\sigma )^{-j}\Vert \nabla ^{k_1-j}u\Vert _{L^{\frac{2m}{k_1-j}}(B_{\sigma '})}\prod _{i=2}^a\Vert \nabla ^{k_i}u\Vert _{L^{\frac{2m}{k_i}}(B_{\sigma '})}\\&\qquad \le C(1-\sigma )^{-j}\Vert u\Vert _{W^{m,2}(B)}^{a}. \end{aligned}$$

• If $1\le j<k_1-m$, then

  $$\begin{aligned}&\Vert \nabla ^{j}\phi \#\nabla ^{k_1-j}u\#\nabla ^{k_2}u\#\cdots \#\nabla ^{k_a}u\Vert _{L^{p}(B)}\\&\qquad \le C(1-\sigma )^{-j}\Vert \nabla ^{k_1-j}u\Vert _{L^{\frac{2mp}{2m-(2m-k_1)p}}(B_{\sigma '})}\prod _{i=2}^a\Vert \nabla ^{k_i}u\Vert _{L^{\frac{2m}{k_i}}(B_{\sigma '})}\\&\qquad \le C(1-\sigma )^{-j}\Vert u\Vert _{W^{2m-j,p}(B_{\sigma '})}\Vert u\Vert _{W^{m,2}(B)}^{a-1}. \end{aligned}$$

  By the interpolation inequality (see [1, Thm. 5.2(1)])

  $$\begin{aligned} \Vert u\Vert _{W^{2m-j,p}(B_{\sigma '})}\le C\left( \Vert \nabla ^{2m-j}u\Vert _{L^{p}(B_{\sigma '})}+\Vert u\Vert _{L^p(B_{\sigma '})}\right) , \end{aligned}$$

  and the boundedness of $\Vert u\Vert _{L^p(B)}$, we conclude that

  $$\begin{aligned}&\Vert \nabla ^{j}\phi \#\nabla ^{k_1-j}u\#\nabla ^{k_2}u\#\cdots \#\nabla ^{k_a}u\Vert _{L^{p}(B)}\\&\quad \le C(1-\sigma )^{-j}\Vert \nabla ^{2m-j}u\Vert _{L^p(B_{\sigma '})}\Vert u\Vert _{W^{m,2}(B)}^{a-1}+C(1-\sigma )^{-j}\Vert u\Vert _{W^{m,2}(B)}^{a-1}. \end{aligned}$$

With the $L^p$ bound of the right-hand side of (4.18) obtained as above, we can apply the $L^p$ estimate to (4.18) to conclude

$$\begin{aligned} \begin{aligned} \Vert \nabla ^{2m}(\phi u)\Vert _{L^p(B)}\le&\, C\sum _{\mathop {\alpha \ne (2m)}\limits ^{|\alpha |=2m}}\Bigg \{ \sum _{j=\max \left\{ k_1-m,1\right\} }^{k_1}(1-\sigma )^{-j}\Vert u\Vert _{W^{m,2}(B)}^{a-1}\\&+\sum _{j=1}^{k_1-m-1}(1-\sigma )^{-j}\Vert u\Vert _{W^{m,2}(B)}^{a-1}\\&+\sum _{j=1}^{k_1-m-1}(1-\sigma )^{-j}\Vert \nabla ^{2m-j}u\Vert _{L^p(B_{\sigma '})}\Vert u\Vert _{W^{m,2}(B)}^{a-1} \Bigg \}\\&+C\sum _{i=1}^{2m}(1-\sigma )^{-i}\Vert \nabla ^{2m-i}u\Vert _{L^p(B_{\sigma '})}. \end{aligned} \end{aligned}$$

(4.20)

Here we have used the small energy condition in Theorem 4.2, which by (4.15) implies that $||u||_{W^{m,2}(B)}$ is small so that the second term in (4.18) are absorbed into the left-hand side.

(4.20) can be further simplified as

$$\begin{aligned} \Vert \nabla ^{2m}u\Vert _{L^p(B_{\sigma })}\le C\sum _{\mathop {\alpha \ne (2m)}\limits ^{|\alpha |=2m}}\sum _{j=1}^{2m}\bigg \{ \Vert u\Vert _{W^{m,2}(B)}^{a-1} +\left( 1+\Vert u\Vert _{W^{m,2}(B)}^{a-1}\right) \Vert \nabla ^{2m-j}u\Vert _{L^p(B_{\sigma '})} \bigg \}(1-\sigma )^{-j}. \nonumber \\ \end{aligned}$$

(4.21)

By setting

$$\begin{aligned} \Phi _j\mathpunct {:}=\sup _{1/2\le \sigma \le 1}(1-\sigma )^j\Vert \nabla ^{j}u\Vert _{L^p(B_\sigma )}, \quad j=0,1,\ldots ,2m, \end{aligned}$$

and noting that $1-\sigma =2(1-\sigma ')$, (4.21) implies

$$\begin{aligned} \Phi _{2m}\le C\sum _{\mathop {\alpha \ne (2m)}\limits ^{|\alpha |=2m}}\sum _{l=0}^{2m-1}\bigg \{ \Vert u\Vert _{W^{m,2}(B)}^{a-1} +\left( 1+\Vert u\Vert _{W^{m,2}(B)}^{a-1}\right) \Phi _l\bigg \}. \end{aligned}$$

(4.22)

Now, we need the following

Claim

(c.f. [6, Thm. 7.27, p. 171ff.]) There exists some constant $C_0$, such that for any ${\bar{\varepsilon }}\le C_0$,

$$\begin{aligned} \Phi _j\le {\bar{\varepsilon }}\Phi _{2m}+C{\bar{\varepsilon }}^{-\frac{j}{2m-j}}\Phi _0,\quad \forall 0\le j\le 2m-1, \end{aligned}$$

holds for some constant C depending on m, j and $C_0$.

In fact, by the definition of $\Phi _j$, for any $s>0$, there exists $\sigma _s\in [1/2,1)$, such that

$$\begin{aligned} \Phi _j\le (1-\sigma _s)^j\Vert \nabla ^ju\Vert _{L^p(B_{\sigma _s})}+s. \end{aligned}$$

For any $u\in W^{2m,p}(\Omega )$, by interpolation inequality (see [1, Theorem 5.2(1)]), there exists a constant $C_0>0$, such that, for all $\varepsilon \le 1$ and any $j=0,1,\ldots ,2m-1$,

$$\begin{aligned} \Vert \nabla ^ju\Vert _{L^p(\Omega )}\le C_0\left( \varepsilon \Vert \nabla ^{2m}u\Vert _{L^p(\Omega )}+\varepsilon ^{-\frac{j}{2m-j}}\Vert u\Vert _{L^p(\Omega )}\right) . \end{aligned}$$

Thus,

$$\begin{aligned} \Phi _j&\le C_0\left( \varepsilon (1-\sigma _s)^j\Vert \nabla ^{2m}u\Vert _{L^p(B_{\sigma _s})}+\varepsilon ^{-\frac{j}{2m-j}}(1-\sigma _s)^j\Vert u\Vert _{L^p(B_{\sigma _s})}\right) +s\\&\le \varepsilon _s\Phi _{2m}+C_0^{1+\frac{j}{2m-j}}(\varepsilon _s)^{-\frac{j}{2m-j}}\Phi _0+s, \end{aligned}$$

where $\varepsilon _s=C_0\varepsilon (1-\sigma _s)^{j-2m}$.

Lastly, for any ${\bar{\varepsilon }}\le C_0$ and any fixed $s>0$ one can take $\varepsilon ={\bar{\varepsilon }}(1-\sigma _s)^{2m-j}/C_0\le 1$, then $\varepsilon _s={\bar{\varepsilon }}$ and the claim follows by taking $s\rightarrow 0$.

Note that (4.15) implies that $\Vert u\Vert _{W^{m,2}(B)}$ are bounded by $E(u;B)\le \varepsilon _0$, which is small by assumption. Applying the claim to (4.22) with small ${\bar{\varepsilon }}$ yields

$$\begin{aligned} \Phi _{2m}\le C\left( \Phi _0+\Vert u\Vert _{W^{m,2}(B)}\right) \le C\left( \Vert \nabla ^mu\Vert _{L^2(B)}+\Vert \nabla u\Vert _{L^{2m}(B)}\right) , \end{aligned}$$

(4.23)

where the last inequality follows from the assumption ${\bar{u}}=0$, the smallness of $\varepsilon _0$ and (4.15). By (4.23) and the interpolation inequality again, we conclude that, for $p=\frac{2m+1}{2m}\in \left( 1,\frac{2m}{2m-1}\right)$,

$$\begin{aligned} \Vert u\Vert _{W^{2m,p}(B_\sigma )}\le C(1-\sigma )^{-2m}\left( \Vert \nabla ^mu\Vert _{L^2(B)}+\Vert \nabla u\Vert _{L^{2m}(B)}\right) . \end{aligned}$$

(4.24)

We will finish the proof by a bootstrap argument. Let us start with $p_0=\frac{2m+1}{2m}<\frac{2m}{2m-1}$. (4.24) implies that (4.12) holds for $p=p_0$.

By the Sobolev embedding $W^{2m,p_0}\hookrightarrow W^{l,\frac{2mp_0}{2m-(2m-l)p_0}}$ and the Hölder inequality,

$$\begin{aligned} \Vert \nabla ^{k_1}u\#\nabla ^{k_2}u\#\cdots \#\nabla ^{k_a}u\Vert _{L^{\frac{1}{1-a(1-1/p_0)}}}&\le C\prod _{i=1}^a\Vert \nabla ^{k_i}u\Vert _{L^{\frac{2mp_0}{2m-(2m-k_i)p_0}}}\\&\le C\Vert u\Vert _{W^{2m,p_0}}^a. \end{aligned}$$

Since $a\ge 2$, we know that

$$\begin{aligned} \frac{1}{1-a\left( 1-\frac{1}{p_0}\right) }\ge \frac{p_0}{2-p_0}=\frac{2m+1}{2m-1}>p_1\mathpunct {:}=\frac{2m}{2m-1}, \end{aligned}$$

(4.25)

thus

$$\begin{aligned} \Vert \nabla ^{k_1}u\#\nabla ^{k_2}u\#\cdots \#\nabla ^{k_a}u\Vert _{L^{p_1}} \le C \Vert u\Vert _{W^{2m,p_0}}^a \end{aligned}$$

(4.26)

In summary, $L^{p_1}$ norm of the right-hand side of (4.17) is bounded by $\Vert u\Vert _{W^{2m,p_0}}$, which implies that (4.12) holds for $p_1$.

Next we do the iteration

$$\begin{aligned} p_{i+1}=\frac{1}{\frac{1}{p_i}-\frac{1}{2m}}=\frac{2m}{2m-i-1},\quad i=1,2,\ldots ,2m-2, \end{aligned}$$

and show that (4.26) holds with $p_0$ and $p_1$ replaced by $p_i$ and $p_{i+1}$, respectively. Thus, the $L^p$-estimate implies that (4.12) holds for $p_{i+1}$.

Note that (4.26) holds for $p_1+\delta _1$, where $\delta _1$ is sufficiently small (see (4.25)). For technical reasons, we will prove the following: Suppose for all $\alpha$ with $|\alpha |=2m\,\text {and}\,\alpha \ne (2m)$,

$$\begin{aligned} \Vert \nabla ^{k_1}u\#\nabla ^{k_2}u\#\cdots \#\nabla ^{k_a}u\Vert _{L^{p_{i+1}+\delta _{i+1}}}\le C\Vert u\Vert _{W^{2m,p_i+\delta _i}}^a, \end{aligned}$$

(4.27)

holds for $i=l\in \left\{ 1,2,\ldots ,2m-3\right\}$ and some $\delta _i>0$, we will show that it also holds for $i=l+1$ and some $\delta _{i+1}$. The exact value of $\left\{ \delta _i\right\}$ does not matter and we only require that they are positive such that (4.28) holds. Choose $\delta _{l}$ sufficiently small, such that

$$\begin{aligned} W^{2m,{\bar{p}}_l}&\hookrightarrow W^{h,\infty },\quad \forall h=0,1,\ldots ,l,\end{aligned}$$

(4.28)

$$\begin{aligned} W^{2m,{\bar{p}}_l}&\hookrightarrow W^{h,\frac{2m{\bar{p}}_l}{2m-(2m-l){\bar{p}}_l}},\quad \forall h=l+1,\ldots ,2m, \end{aligned}$$

(4.29)

where ${\bar{p}}_l=p_l+\delta _l$. To illustrate the idea, let us assume that the multi-indices $\alpha$ in (4.27) satisfy

$$\begin{aligned} k_1\ge k_2\ge \cdots \ge k_{a-j_1-j_2\cdots -j_l}&>k_{a-j_1-j_2\cdots -j_l+1}=\cdots =k_{a-j_1-j_2\cdots -j_{l-1}}=l\\&>\cdots \\&>k_{a-j_1-j_2+1}=\cdots =k_{a-j_1}=2\\&>k_{a-j_1+1}=\cdots =k_{a}=1. \end{aligned}$$

Setting $j=j_1+j_2\cdots +j_{l}$, by Hölder and Sobolev inequality

$$\begin{aligned}&\Vert \nabla ^{k_1}u\#\cdots \#\nabla ^{k_a}u\Vert _{L^p}\\&\qquad \le C\Vert \nabla ^{k_1}u\#\cdots \#\nabla ^{k_{a-j}}u\Vert _{L^p} \prod _{h=1}^{j_{l}}\Vert \nabla ^{k_{a-j+h}}u\Vert _{L^\infty }\cdots \prod _{h=1}^{j_1}\Vert \nabla ^{k_{a-j_1+h}}u\Vert _{L^\infty }\\&\qquad \le C\Vert u\Vert _{W^{2m, {\bar{p}}_l}}^{j} \prod _{h=1}^{a-j}\Vert \nabla ^{k_h}u\Vert _{L^{\frac{2m{\bar{p}}_l}{2m-(2m-k_h){\bar{p}}_l}}},\quad \text {(by (5.17))}\\&\qquad \le C\Vert u\Vert _{W^{2m,{\bar{p}}_l}}^a,\quad \text {(by (5.18))} \end{aligned}$$

where

$$\begin{aligned} p\le \frac{1}{1-(a-j)\left( 1-\frac{1}{{\bar{p}}_l} \right) -\frac{j_1+2j_2+\cdots +lj_l}{2m}} \end{aligned}$$

which attains its minimum at $j_l=\cdots =j_2=0, j_1=1, a=2$, i.e.,

$$\begin{aligned} p\ge \frac{1}{\frac{1}{{\bar{p}}_l}-\frac{1}{2m}}\mathpunct {:}={\bar{p}}_{l+1}>p_{l+1}. \end{aligned}$$

Therefore, (4.24) holds for $i=l+1$.

As a conclusion of the above iteration, we have shown that (4.12) holds for some ${\bar{p}}>2m=p_{2m-1}$. Now for general $p>1$, note that

$$\begin{aligned} W^{2m,{\bar{p}}}\hookrightarrow W^{h,\infty },\quad \forall h=0,1,\ldots ,2m-1. \end{aligned}$$

Thus,

$$\begin{aligned} \Vert \nabla ^{k_1}u\#\nabla ^{k_2}u\#\cdots \#\nabla ^{k_a}u\Vert _{L^p}&\le C\Vert \nabla ^{k_1}u\#\nabla ^{k_2}u\#\cdots \#\nabla ^{k_a}u\Vert _{L^\infty } \le C\Vert u\Vert _{W^{2m,{\bar{p}}}}^a, \end{aligned}$$

which implies that (4.12) holds for p and we finish the first part of Theorem 4.2.

Finally, (4.13) follows from (4.12) and the standard bootstrap argument. $\square$

Appendix 2: Linearized polyharmonic map equation and some higher order estimates

In Sects. 3 and  4, by using either the three-circle lemma (Lemma 2.1) or the Pohozaev-type argument, we have proved the decay of the $L^2$ norm of some derivative of $u_i$ ($X_k u_i$ and $\partial _t u_i$, respectively) along the neck. In this section, we provide a lemma which improves the decay of $L^2$ norm to the decay of pointwise higher order norm. The key to the proof is the observation that $X_k u_i$ and $\partial _t u_i=\rho \partial _\rho u_i$ satisfy a homogeneous linear elliptic system with nice coefficients.

Let u be a smooth polyharmonic map defined on $B_4{\setminus } B_1$. Recall the Euler–Lagrange equation reads

$$\begin{aligned} \Delta ^m u = \sum _{\mathop {\alpha \ne (2m)}\limits ^{|\alpha |=2m}} a_\alpha (u) \nabla ^{k_1} u \# \cdots \# \nabla ^{k_a}u. \end{aligned}$$

Moreover, we assume that

$$\begin{aligned} \max _{B_4{\setminus } B_1} \left| \nabla ^l u\right| \le 1,\quad \forall l=1,2,\ldots ,2m-1, \end{aligned}$$

(4.30)

which holds for each segment of neck due to the Ding-Tian’s reduction (see [4]) and $\varepsilon$-regularity theorem (Theorem 4.2).

Lemma 4.3

Suppose u is a polyharmonic map defined on $B_4{\setminus } B_1$ satisfying (4.30) as above. Then,

$$\begin{aligned} \max _{B_3{\setminus } B_2} \left| \nabla ^l ( (\rho \partial _\rho ) u)\right| ^2 \le C \int _{B_4{\setminus } B_1} \left| \rho \partial _\rho u\right| ^2 \mathrm{d}x \end{aligned}$$

(4.31)

and

$$\begin{aligned} \max _{B_3{\setminus } B_2} \left| \nabla ^l ( X_k u)\right| ^2 \le C \int _{B_4{\setminus } B_1} \left| X_k u\right| ^2 \mathrm{d}x \end{aligned}$$

(4.32)

for $k=1,2,\cdots ,m(2m-1)$.

Proof

The proof follows from well-known elliptic estimates if we can show that $X_ku$ and $(\rho \partial _\rho ) u$ satisfy a nice linear equation so that we can apply linear estimates.

We make use of the fact that the polyharmonic map equation is invariant under the one parameter group generated by $\rho \partial _\rho$ or any one of the Killing vector fields $\left\{ X_k\right\}$. More precisely, let $\psi _s$ be such a one parameter group and $u_s= u\circ \psi _s$. If u is m-polyharmonic map, then so is $u_s$. Therefore, $u_s$ satisfies (c.f. (4.16))

$$\begin{aligned} \Delta ^m u_s = \sum _{\mathop {\alpha \ne (2m)}\limits ^{|\alpha |=2m}} a_\alpha (u_s) \nabla ^{k_1} u_s \# \cdots \# \nabla ^{k_a} u_s. \end{aligned}$$

Taking s-derivative at $s=0$ and denoting $\frac{du_s}{ds}|_{s=0}$ by h gives

$$\begin{aligned} \Delta ^m h= & {} \sum _{\mathop {\alpha \ne (2m)}\limits ^{|\alpha |=2m}} D a_\alpha (u) \nabla ^{k_1} u \# \cdots \# \nabla ^{k_a} u \cdot h \\&+ \sum _{\mathop {\alpha \ne (2m)}\limits ^{|\alpha |=2m}} a_\alpha (u) \left( \nabla ^{k_1} h \# \cdots \# \nabla ^{k_a} u + \cdots + \nabla ^{k_1} u \# \cdots \# \nabla ^{k_a}h \right) . \end{aligned}$$

This is a linear elliptic system of h, whose coefficients are all good by (4.30).

Finally, we notice that h can be either $(\rho \partial _\rho ) u$ or $X_ku$ in the above computation, depending on the choice of the one parameter group $\psi _s$. $\square$