Boundary regularity for minimizing biharmonic maps

Abstract

We prove full boundary regularity for minimizing biharmonic maps with smooth Dirichlet boundary conditions. Our result, similarly as in the case of harmonic maps, is based on the nonexistence of nonconstant boundary tangent maps. With the help of recently derivated boundary monotonicity formula for minimizing biharmonic maps by Altuntas we prove compactness at the boundary following Scheven’s interior argument. Then we combine those results with the conditional partial boundary regularity result for stationary biharmonic maps by Gong–Lamm–Wang.

Appendix A

Lemma A.1

There is a constant C depending only on m such that for any $0<r<R$ and any map $u\in W^{2,2}(B_R^+,\mathbb {R}^\ell )$ with vanishing $W^{2,2}$ trace on $T_R = \{x\in B_R:x_m=0\}$ we have

$$\begin{aligned} \int _{B_r^+}|\nabla ^2 u|^2 \,dx\le 2 \int _{B_R^+} \left( {|\Delta u|^2 + \frac{C}{(R-r)^2}|\nabla u|^2}\right) \,dx. \end{aligned}$$

(A.1)

Proof

The proof is exactly as in [21, Lemma A.1]. We choose the same cut-off function $\eta \in C_c^\infty (B_R,[0,1])$ such that $\eta \equiv 1$ on $B_r$ and $|\nabla \eta |<\frac{C}{(R-r)}$. If we assume that $u\in C^\infty (B_R^+,\mathbb {R}^l)$ and integrate twice by parts the following integral

$$\begin{aligned} \int _{B_R^+} \eta ^4|\Delta |^2 \,dx, \end{aligned}$$

the boundary term will vanish on the flat part of $\partial B_R^+$, because the $W^{2,2}$ trace of u vanishes there. It will also vanish on the curved part of the boundary, for $\eta$ vanishes there. Thus,

$$\begin{aligned} \int _{B_R^+}\eta ^4 u_{x_i x_i}u_{x_j x_j}\,dx&= - 4\int _{B_R^+}\eta ^3\eta _{x_j} u_{x_i x_i}u_{x_j}\,dx- \int _{B_R^+}\eta ^4 u_{x_i x_i x_j}u_{x_j}\,dx\\&= - 4\int _{B_R^+}\eta ^3\eta _{x_j} u_{x_i x_i}u_{x_j}\,dx+ 4\int _{B_R^+}\eta ^3\eta _{x_i} u_{x_i x_j}u_{x_j}\,dx\\&\quad \,\, + \int _{B_R^+}\eta ^4 u_{x_i x_j}u_{x_i x_j}\,dx. \end{aligned}$$

Hence,

$$\begin{aligned} \int _{B_R^+}\eta ^4 |\nabla ^2 u|^2 \,dx&\le \int _{B_R^+}\eta ^4 |\Delta u|^2 \,dx+ C\int _{B_R^+}\eta ^3|\nabla \eta ||\nabla u||\nabla ^2 u|\,dx\\&\le \int _{B_R^+}\eta ^4 |\Delta u|^2 \,dx+ \frac{1}{2} \int _{B_R^+}\eta ^4 |\nabla ^2 u|^2\,dx\\&\quad + \frac{C}{(R-r)^2}\int _{B_R^+}\eta ^2|\nabla u|^2 \,dx. \end{aligned}$$

The desired inequality for smooth u follows by subtracting $\frac{1}{2} \int _{B_R^+}\eta ^4 |\nabla ^2 u|^2\,dx$ from both sides. Now an approximation argument yields the the same argument for $W^{2,2}$ maps. $\square$

The next Lemma shows that by boundary monotonicity formula a bound in $W^{2,2}$ implies a bound in the Morrey space $L^{2,m-4}$. The proof is almost identical to the proof in the interior case, but as the boundary monotonicity formula yields an additional term we sketch the proof below.

Lemma A.2

Let $u\in W^{2,2}(B^+_4,\mathcal {N})$ be a minimizing biharmonic map with boundary value $\varphi$ as in (1.5) and let$\left\| {u-\varphi }\right\| _{W^{2,2}(B^+)}<\infty$. Let $\widetilde{u}$ be the reflection of $u-\varphi$ given in 3.3, then

$$\begin{aligned} \sup _{y\in B,\ \rho <1} \rho ^{4-m}\int _{B_\rho (y)}|\nabla ^2 \widetilde{u}|^2 \,dx\le C \int _{B_2}|\nabla ^2 \widetilde{u}|^2 \,dx+ \widetilde{C}, \end{aligned}$$

(A.2)

for constants $C=C(m)$ and $\widetilde{C} = \widetilde{C}(m,\mathcal {N})$.

Proof of Lemma A.2

We give the necessary modification of [21, Lemma A.2].

We note that since u satisfies the boundary monotonicity formula (2.6) we have for $\widetilde{u}$ and $a\in T_1$ the following

$$\begin{aligned}&\rho ^{4-m}\int _{B_\rho (a)}|\nabla ^2 \widetilde{u}|^2 \,dx+ Ce^{C\rho } R^+_u(a,\rho )\nonumber \\&\quad \le C\left( {e^{Cr}r^{4-m}\int _{B_r(a)}|\nabla ^2 \widetilde{u}|^2 \,dx+ e^{Cr} R^+_u(a,r) + Cre^{Cr}}\right) . \end{aligned}$$

(A.3)

Let $0<s<1/8$ be given. By Fubini theorem we may choose good radii $\rho <r$ with $s\le \rho \le 2s<\frac{1}{2}\le r\le 1$ such that

$$\begin{aligned} \rho ^{3-m}\int _{B_\rho (a)}|\nabla \widetilde{u}|^2 \,d\mathcal {H}^{m-1}&\le C s^{2-m}\int _{B_{2s(a)}}|\nabla \widetilde{u}|^2 \, dx;\\ \rho ^{5-m}\int _{B_\rho (a)}|\nabla ^2 \widetilde{u}|^2 \,d\mathcal {H}^{m-1}&\le C s^{4-m}\int _{B_{2s(a)}}|\nabla ^2 \widetilde{u}|^2 \, dx;\\ \int _{B_r(a)}\left( {|\nabla ^2 \widetilde{u}|^2 + |\nabla \widetilde{u}|^2}\right) \,d\mathcal {H}^{m-1}&\le C\int _{B_1(a)}\left( {|\nabla ^2 \widetilde{u}|^2 + |\nabla \widetilde{u}|^2}\right) \, dx, \end{aligned}$$

where the constant C depends only on the dimension m.

One can easily observe that

$$\begin{aligned} \left| R^+_{u,\varphi }(a,\tau )\right|&\le C\tau ^{4-m}\int _{\partial B^+_\tau (a)}\left( {|\nabla ^2 u||\nabla u| + \frac{1}{\tau }|\nabla u|^2}\right) \,d\mathcal {H}^{m-1}\\&\le C\tau ^{4-m}\int _{\partial B_\tau (a)}\left( {|\nabla ^2 \widetilde{u}||\nabla \widetilde{u}| + \frac{1}{\tau }|\nabla \widetilde{u}|^2}\right) \,d\mathcal {H}^{m-1}. \end{aligned}$$

Combining this observation with (A.3) we get

$$\begin{aligned} \rho ^{4-m}\int _{B_\rho (a)}|\nabla ^2 \widetilde{u}|^2 \,dx&\le Ce^{Cr}r^{4-m}\int _{B_r(a)}|\nabla ^2 \widetilde{u}|^2 \,dx+ Cre^{Cr}\\&\quad + C\rho ^{4-m}\int _{\partial B_\rho (a)}\left( {|\nabla ^2 \widetilde{u}||\nabla \widetilde{u}| + \frac{1}{\rho }|\nabla \widetilde{u}|^2}\right) \,d\mathcal {H}^{m-1}\\&\quad + C e^{Cr}r^{4-m}\int _{\partial B_r(a)}\left( {|\nabla ^2 \widetilde{u}||\nabla \widetilde{u}| + \frac{1}{r}|\nabla \widetilde{u}|^2}\right) \,d\mathcal {H}^{m-1}.\\ \end{aligned}$$

Thus, since $s<\rho <2s$ and by Young’s inequality with $\epsilon$

$$\begin{aligned} s^{4-m}\int _{B_s(a)}|\nabla ^2 \widetilde{u}|^2 \,dx&\le C\rho ^{4-m}\int _{B_\rho (a)}|\nabla ^2 \widetilde{u}|^2 \,dx\nonumber \\&\le \frac{1}{4} (2s)^{4-m} \int _{B_{2s}(a)}|\nabla ^2 \widetilde{u}|^2\ dx + C s^{2-m}\int _{B_{2s}(a)}|\nabla \widetilde{u}|^2 \,dx\nonumber \\&\quad + C\int _{B_1(a)}|\nabla ^2 \widetilde{u}|^2 \,dx+ C\int _{B_1(a)}|\nabla \widetilde{u}|^2 \,dx+ C. \end{aligned}$$

(A.4)

Next, we proceed exactly as in [21]. Observe that by Nirenberg’s interpolation inequality

$$\begin{aligned} \Vert \nabla f\Vert _{L^4(\Omega )}^2 \le C(\Omega ) \Vert f\Vert _{L^\infty (\Omega )}\Vert f\Vert _{W^{2,2}(\Omega )} \end{aligned}$$

we have after a few transformations

$$\begin{aligned} C\tau ^{2-m}\int _{B_\tau (y)}|\nabla \widetilde{u}|^2 \,dx\le \frac{1}{4} \tau ^{4-m}\int _{B_{\tau }(y)}|\nabla \widetilde{u}|^2\,dx+ \widetilde{C} \end{aligned}$$

(A.5)

where $\widetilde{C}$ is a constant dependent on the target manifold $\mathcal {N}$. Applying (A.5) into (A.4) for $\tau =1$ and $\tau = 2s$, denoting $\widehat{H}(\tau ):= \tau ^{4-m}\int _{B_\tau (y)}|\nabla ^2 \widetilde{u}|^2 \,dx$, we arrive at

$$\begin{aligned} \widehat{H}(s)\le \frac{1}{2} \widehat{H}(2s) + C\widehat{H}(1) + \widetilde{C} \end{aligned}$$

for all $0<s<\frac{1}{4}$.

Thus, for all small $\sigma >0$

$$\begin{aligned} \sup _{\sigma<s<1} \widehat{H}(s) \le \sup _{\sigma<s<1/4} \widehat{H}(s) + C\widehat{H}(1) \le \frac{1}{2} \sup _{\sigma<s<1/4} \widehat{H}(2s)+ C\widehat{H}(1) + \widetilde{C}. \end{aligned}$$

Since $\sigma >0$ the term $\frac{1}{2} \sup _{\sigma<s<1/4} \widehat{H}({2}s)$ is finite and can be absorbed by the left hand side of the inequality giving

$$\begin{aligned} \sup _{\sigma<s<1}\widehat{H}(\rho )\le C\widehat{H}(1) + \widetilde{C}. \end{aligned}$$

The estimate is independent of $\sigma >0$ and thus the claimed inequality follows. $\square$

Remark A.3

In the last proof we did not need a higher order reflection. An odd reflection is enough to ensure that if $u-\varphi \in W^{2,2}_0(B^+,\mathbb {R}^\ell )$, then the reflected map is in $W^{2,2}(B,\mathbb {R}^\ell )$.