Blow-up analysis and boundary regularity for variationally biharmonic maps

TL;DR

This paper studies the boundary regularity of variationally biharmonic maps, showing that the absence of certain biharmonic spheres in the target guarantees smoothness up to the boundary.

Contribution

It establishes boundary regularity results for variationally biharmonic maps by linking potential singularities to the existence of biharmonic spheres in the target manifold.

Findings

Weak convergence of maps is obstructed only by biharmonic spheres or half-spheres.

Boundary regularity holds if such biharmonic spheres do not exist in the target.

Identifies conditions under which boundary regularity is achieved.

Abstract

We consider critical points $u:\Omega\to N$ of the bi-energy \[ \int_\Omega |\Delta u|^2\,d x, \] where $\Omega\subset\mathbb{R}^m$ is a bounded smooth domain of dimension $m\ge 5$ and $N\subset\mathbb{R}^L$ a compact submanifold without boundary. More precisely, we consider variationally biharmonic maps $u\in W^{2,2}(\Omega,N)$, which are defined as critical points of the bi-energy that satisfy a certain stationarity condition up to the boundary. For weakly convergent sequences of variationally biharmonic maps, we demonstrate that the only obstruction that can prevent the strong compactness up to the boundary is the presence of certain non-constant biharmonic $4$-spheres or $4$-halfspheres in the target manifold. As an application, we deduce full boundary regularity of variationally biharmonic maps provided such spheres do not exist.

Equations

$${\int }_{\Omega }|\Delta u{|}^2\mathrm{d}x,$$

$$E_1(u):={\int }_{\Omega }|Du{|}^2\mathrm{d}x.$$

$$E_2(u):={\int }_{\Omega }|\Delta u{|}^2\mathrm{d}x.$$

$$E_2(u):={\int }_{\Omega }|\Delta u{|}^2\mathrm{d}x\text{mbox}foru\in W^{2,2}(\Omega ,N).$$

$${\int }_{\Omega }\Delta u\cdot \Delta Vdx=0$$

$$\Delta^{2}u=\Delta\big{(}A(u)(Du\otimes Du)\big{)}+2\mathrm{div}\,\big{(}D(\Pi(u))\cdot\Delta u\big{)}-\Delta(\Pi(u))\cdot\Delta u$$

$$\displaystyle\int_{\Omega}\big{(}4\Delta u\cdot D^{2}uD\xi+2\Delta u\cdot Du\,\Delta\xi-|\Delta u|^{2}\mathrm{div}\,\xi\big{)}\,\mathrm{d}x=0$$

$$(u,Du)=(g,Dg)\qquad\mbox{on $\Gamma$ in the sense of traces,}$$

$$\frac{d}{ds}\bigg{|}_{s=0}E_{2}(u_{s})=0$$

$$P_{S^{4}}(u):=\int_{S^{4}}\big{[}|\Delta_{S}u|^{2}+2|Du|^{2}\big{]}\mathrm{d}x\qquad\mbox{for }u\in W^{2,2}(S^{k},N),$$

$${\Delta }_S^{2}u(x)-2{\Delta }_{S}u(x)\perp T_{u(x)}N$$

$$\underset{i\in \text{mathds}N}{\sup }\Vert u_i{\Vert }_{W^{2,2}(B_4^{+})}<\infty \text{mbox}and\underset{i\in \text{mathds}N}{\sup }\Vert g_i{\Vert }_{C^{4,\alpha }(B_4^{+})}<\infty ,$$

$$u_{i}\to u\qquad\mbox{in $W^{2,2}(B_{1}^{+},{\mathds{R}}^{L})$, as $i\to\infty$.}$$

$$u_i\in C^{\infty }(B_1^{4,+},N),u_i(x):=u({\lambda }_{i}x)\text{mbox}forx\in B_1^{4,+}\subset {\text{mathds}R}_{+}^4,$$

$$\underset{i\in \text{mathds}N}{\sup }{\int }_{B_1^{4,+}}|\Delta u_i{|}^{2}dx=\underset{i\in \text{mathds}N}{\sup }{\int }_{B_{{\lambda }_i}^{4,+}}|\Delta u{|}^{2}dx={\int }_{{\text{mathds}R}_{+}^4}|\Delta u{|}^{2}dx<\infty ,$$

$${\int }_{B_1^{4,+}}|\Delta u_i(x){|}^2\phi (x)\mathrm{d}x={\int }_{B_{{\lambda }_i}^{4,+}}|\Delta u(y){|}^2\phi ({\lambda }_i^{-1}y)\mathrm{d}y\to \phi (0){\int }_{{\text{mathds}R}_{+}^4}|\Delta u{|}^2\mathrm{d}y,$$

$$\Vert f{\Vert }_{L^{2,\lambda }(B_2^{+})}^2:=\underset{B_{\rho }^{+}(y)\subset B_2^{+}}{\sup }\frac{1}{{\rho }^{\lambda }}{\int }_{B_{\rho }^{+}(y)}|f{|}^2\mathrm{d}x.$$

$$\mathrm{sing}(u):=\overline{\Omega }\setminus \left\{x\in \overline{\Omega }:u\in C^{\infty }(B_{\rho }(x)\cap \overline{\Omega },{\text{mathds}R}^L)\text{mbox}forsome\rho >0\right\}.$$

$$\displaystyle\int_{\Omega}\big{(}4\Delta u\cdot D^{2}uD\xi+2\Delta u\cdot Du\,\Delta\xi-|\Delta u|^{2}\mathrm{div}\,\xi\big{)}\,\mathrm{d}x$$

$$\displaystyle\quad=\int_{\Omega}2\Delta u\cdot\Delta\big{[}\Pi(u)(Dg\,\xi)\big{]}\,\mathrm{d}x,$$

$$\displaystyle\Phi_{u}(a;\rho)+4\int_{B_{r}^{+}(a)\setminus B_{\rho}^{+}(a)}e^{\chi|x-a|}\bigg{(}\frac{|D\partial_{X}u|^{2}}{|x-a|^{m-2}}+(m-2)\frac{|\partial_{X}u|^{2}}{|x-a|^{m}}\bigg{)}\,\mathrm{d}x$$

$$\le {\Phi }_u(a;r)+K{\Psi }_u(a;\rho ,r),$$

$${\Phi }_u(a;r)$$

$$\displaystyle\qquad+e^{\chi r}r^{3-m}\int_{S_{r}^{+}(a)}\big{(}\partial_{X}|Du|^{2}+4|Du|^{2}-4r^{-2}|\partial_{X}u|^{2}\big{)}\,d\mathcal{H}^{m-1},$$

$${\Psi }_u(a;\rho ,r)$$

$$+{\int }_{S_r^{+}(a)\cup S_{\rho }^{+}(a)}\frac{|D^{2}u{|}^2}{|x-a{|}^{m-6}}d{\mathcal{H}}^{m-1}.$$

$$\underset{B_{\rho }^{+}(y)\subset B_{R/2}^{+}(a)}{\sup }$$

$$\displaystyle\leq c_{1}R^{4-m}\int_{B_{R}^{+}(a)}\big{(}|\Delta u|^{2}+R^{-2}|Du|^{2}\big{)}\,\mathrm{d}x+c_{2}R,$$

$${\tilde{\Phi }}_u(a;\rho ):=e^{\chi \rho }{\rho }^{4-m}{\int }_{B_{\rho }^{+}(a)}|\Delta u{|}^2\mathrm{d}x+e^{\chi \rho }{\rho }^{3-m}{\int }_{S_{\rho }^{+}(a)}|Du{|}^2\mathrm{d}{\mathcal{H}}^{m-1},$$

$${\tilde{\Phi }}_u(a;\rho )-{\Phi }_u(a;\rho )$$

$$\displaystyle\qquad=e^{\chi\rho}\rho^{3-m}\int_{S_{\rho}^{+}(a)}\big{(}4\rho^{-2}|\partial_{X}u|^{2}-\partial_{X}|Du|^{2}-3|Du|^{2}\big{)}\mathrm{d}\mathcal{H}^{m-1}$$

$$\displaystyle\qquad=e^{\chi\rho}\rho^{3-m}\int_{S_{\rho}^{+}(a)}\big{(}4\rho^{-2}|\partial_{X}u|^{2}-2D(\partial_{X}u)\cdot Du-|Du|^{2}\big{)}\mathrm{d}\mathcal{H}^{m-1}$$

$$\displaystyle\qquad\leq e^{\chi\rho}\rho^{3-m}\int_{S_{\rho}^{+}(a)}\big{(}4\rho^{-2}|\partial_{X}u|^{2}+|D\partial_{X}u|^{2}\big{)}\mathrm{d}\mathcal{H}^{m-1},$$

$$\text{vbox}-{\int }_{2s}^{4s}{\tilde{\Phi }}_u(a;\rho )\mathrm{d}\rho$$

Full text

[TABLE]

where $\Omega\subset{\mathds{R}}^{m}$ is a bounded smooth domain of dimension $m\geq 5$ and $N\subset{\mathds{R}}^{L}$ a compact submanifold without boundary. More precisely, we consider variationally biharmonic maps $u\in W^{2,2}(\Omega,N)$, which are defined as critical points of the bi-energy that satisfy a certain stationarity condition up to the boundary. For weakly convergent sequences of variationally biharmonic maps, we demonstrate that the only obstruction that can prevent the strong compactness up to the boundary is the presence of certain non-constant biharmonic $4$-spheres or $4$-halfspheres in the target manifold. As an application, we deduce full boundary regularity of variationally biharmonic maps provided such spheres do not exist.

1. Introduction and statement of the results

Biharmonic maps are a higher order variant of harmonic maps $u\in C^{\infty}(\Omega,N)$ into a Riemannian manifold $N\subset{\mathds{R}}^{L}$, which are defined as critical points of the Dirichlet energy

[TABLE]

Analogously, we call a map $u\in C^{\infty}(\Omega,N)$ biharmonic if it is a critical point of the bi-energy

[TABLE]

More generally, maps $u\in W^{2,2}(\Omega,N)$ that satisfy the Euler-Lagrange equation of $E_{2}$ in the weak sense are called weakly biharmonic, and the class of weakly harmonic maps is defined accordingly. The analytical and geometric properties of harmonic maps have been extensively studied over the last decades and are quite well understood. We refer to [6] for an overview over the classical theory. The theory of biharmonic maps, however, is not yet developed up to the same level as the one of harmonic maps. In the present article, we analyse the behaviour of biharmonic maps at the boundary and investigate the questions of compactness properties and regularity up to the boundary.

Before discussing the state of the regularity theory for biharmonic maps, let us briefly recall some of the main results on regularity of harmonic maps. For minimizing harmonic maps, i.e. minimizers of the Dirichlet energy in a given Dirichlet class, Schoen & Uhlenbeck proved that the singular set can have at most Hausdorff-dimension $m-3$, see [24]. An alternative proof was later given by Luckhaus [11]. Moreover, in [25], Schoen & Uhlenbeck were even able to prove full regularity in a neighbourhood of the boundary. For harmonic maps that are not minimizing, only slightly weaker results are known. First of all, no regularity results can be derived in super-critical dimensions $m>2$ for weakly harmonic maps that do not satisfy a certain energy monotonicity formula, cf. [17]. For the slightly smaller class of stationary harmonic maps, however, Bethuel [3] established that the singular set has vanishing $(m-2)$-dimensional Hausdorff measure, see also [18, 19] for an alternative proof. The reduction of the dimension to the upper bound $m-3$ as in the case of minimizers is not known in the general situation. On a technical level, the reason is that weakly convergent sequences of stationary harmonic maps may not have a strongly convergent subsequence [10, Example 1.1], differently from the case of minimizing harmonic maps [24, 11]. Therefore, it is not possible to derive the dimension bound for the singular set by means of Federer’s dimension reduction principle. However, a deep result by Lin [10] states that this lack of compactness can occur if and only if the target manifold contains a non-constant smooth harmonic $2$-sphere $v:S^{2}\to N$. As a consequence, under the assumption that the target manifold does not carry any non-trivial harmonic $2$-spheres, it is possible to prove full regularity in the neighbourhood of the boundary also for more general critical points and not only for minimizers [10, 20].

The regularity of biharmonic maps was first investigated by Chang, Wang & Yang [5], see also [28, 29, 30], with the result that for any stationary biharmonic map, the set of interior singular points is negligible with respect to the $(m-4)$-dimensional Hausdorff measure. In the case of minimizing biharmonic maps, the dimension of the interior singular set was further reduced to at most $m-5$ by the second author [21]. The latter article also contains results for stationary biharmonic maps under the assumption that the target manifold does not carry any non-constant Paneitz-biharmonic $4$-spheres (cf. Definition 1.4), which turns out to be the analogue of the condition found by Lin [10] in the harmonic map case. As for harmonic maps, an indispensable tool for all mentioned partial regularity results in super-critical dimensions $m>4$ is an energy monotonicity formula. For biharmonic maps, this formula was derived in the interior case by Angelsberg [2]. In the boundary situation, however, the question for the corresponding monotonicity formula remained open for some time. In fact, since such a formula was unknown, the first results on partial boundary regularity [7] and full boundary regularity for minimizers [13] had to impose this monotonicity property as an additional assumption. This gap in the theory has been closed by the first author [1], who provided a suitable boundary monotonicity formula and thereby completed the mentioned results from [7, 13].

The present article now is concerned with the question whether full boundary regularity can also be derived for biharmonic maps that are not minimizing, but only critical points of the bi-energy. The suitable notion of critical point in the boundary situation is that of a variationally biharmonic map, see Definition 1.2. This notion, which is slightly stronger than that of a stationary biharmonic map, has been introduced in [20] in the harmonic map case and allows in particular to use any variation of the domain that keeps the boundary values fixed. Our first main result is a compactness property for sequences of variationally biharmonic maps. For the proof, we adapt the strategy from [21], which are in turn based on [10], to the boundary case. The arguments consist in an intricate blow-up analysis of the defect measure, which detects a possible lack of strong convergence. We achieve basically the analogous result as in the harmonic map case, with the only exception that additionally to the non-existence of non-trivial Paneitz-biharmonic $4$-spheres, we also need to exclude the existence of non-constant Paneitz-biharmonic $4$-halfspheres with constant boundary values. The reason is that the non-existence proof of harmonic $2$-halfspheres from [9] does not seem to carry over to the higher order case. However, it seems to be plausible that whenever it is possible to exclude nontrivial Paneitz-biharmonic $4$-spheres, the same arguments will also yield the nonexistence of the corresponding halfspheres. An example of this principle is given in Proposition 4.4.

This compactness property is the prerequisite for our second main result, which ensures the full boundary regularity of variationally biharmonic maps under the assumption that no biharmonic $4$-spheres and $4$-halfspheres as above exist in the target manifold. For the proof, we follow Federer’s dimension reduction argument and analyse tangent maps of variationally biharmonic maps in singular boundary points. For the construction of the tangent maps, it is crucial to have the strong convergence properties from our first main result. Since it is possible to show that all tangent maps necessarily have to be constant, we can deduce that singular boundary points do not exist.

Next, we specify our assumptions and state our main results.

1.1. Variationally biharmonic maps

Let $\Omega\subset{\mathds{R}}^{m}$ be a bounded domain of dimension $m\geq 5$. We prescribe Dirichlet boundary data on a boundary part $\Gamma\subset\partial\Omega$, where the boundary datum is given in form of a map $g\in C^{3}(\Gamma_{\delta},N)$ defined on a neighborhood $\Gamma_{\delta}:=\{x\in\overline{\Omega}\,:\,\mathrm{dist}(x,\Gamma)<\delta\}$ of $\Gamma$. We consider critical points of the bi-energy

[TABLE]

The following notion of weakly biharmonic maps can be derived by considering variations of the type $u_{s}(x):=\pi_{N}(u(x)+sV(x))$ with the nearest-point retraction $\pi_{N}$ onto $N$.

Definition 1.1**.**

A map $u\in W^{2,2}(\Omega,N)$ is called weakly biharmonic iff

[TABLE]

holds true for every vector field $V\in W^{2,2}_{0}\cap L^{\infty}(\Omega,{\mathds{R}}^{L})$ that is tangential along $u$ in the sense $V(x)\in T_{u(x)}N$ for a.e. $x\in\Omega$.

In other words, a weakly biharmonic map is characterized by the fact that $\Delta^{2}u\perp T_{u}N$ holds in the weak sense. This is equivalent to the differential equation

[TABLE]

in the distributional sense, where $A(u)$ denotes the second fundamental form of $N\subset{\mathds{R}}^{L}$ and $\Pi(u(x)):{\mathds{R}}^{L}\to T_{u(x)}N$ is the orthogonal projection onto the tangent space at $u(x)\in N$. For a detailed proof, we refer to [30, Prop. 2.2]. Classical solutions $u\in C^{4}(\Omega,N)$ of (1) are called biharmonic maps.

As mentioned above, in super-critical dimensions $m\geq 5$ a monotonicity formula is crucial for the derivation of regularity results. Since such a formula can not be expected to hold for general weakly biharmonic maps, we have to consider a stronger notion of biharmonicity. Considering variations of the type $u_{s}(x)=u(x+s\xi(x))$ leads to the following notion of biharmonic maps.

Definition 1.2**.**

A map $u\in W^{2,2}(\Omega,N)$ is called stationary biharmonic iff it is weakly biharmonic in $\Omega$ and satisfies the differential equation

[TABLE]

for every $\xi\in C^{\infty}_{0}(\Omega,{\mathds{R}}^{m})$.

However, it turns out that this notion is still not sufficient for the treatment of the Dirichlet problem, since the differential equation (2) only contains information on interior properties of solutions. Therefore, we rely on the following notion of biharmonic maps that is adapted to the Dirichlet boundary problem.

Definition 1.3**.**

A map $u\in W^{2,2}(\Omega,N)$ is called variationally biharmonic with respect to the Dirichlet datum $g$ on $\Gamma$ if

[TABLE]

and if

[TABLE]

holds true for every variation $u_{s}\in W^{2,2}(\Omega,N)$, $s\in(-\varepsilon,\varepsilon)$, for which the above derivative exists, which satisfies $u_{0}=u$, the boundary condition (3) with $u_{s}$ in place of $u$, and $u_{s}=u$ a.e. on $\Omega\setminus K$ for some compact set $K\subset\Omega\cup\Gamma$.

1.1.1. Paneitz-biharmonic maps

For maps on the $k$-dimensional upper halfsphere $S^{k}_{+}$, there are corresponding notions of biharmonicity. For the purposes of the present article, it suffices to consider the case $k=4$. We prescribe boundary values in form of a map $g\in C^{3}(U_{\delta},N)$, where we abbreviated $U_{\delta}:=S^{4}_{+}\cap({\mathds{R}}^{4}\times[0,\delta))$. For maps defined on the $4$-sphere $S^{4}$, we define the Paneitz-bi-energy by

[TABLE]

with the Laplace-Beltrami operator $\Delta_{S}$ on $S^{4}$. Analogously, we define $P_{S^{4}_{+}}(u)$ for any $u\in W^{2,2}(S^{4}_{+},N)$. The Euler-Lagrange operator of this functional is given by the Paneitz-operator $\mathcal{P}u:=\Delta_{S}^{2}u-2\Delta_{S}u$ on $S^{4}$, which plays an important role in comformal geometry. In particular, well-known properties of the Paneitz operator imply that the Paneitz-bi-energy $P_{S^{4}}$ is conformally invariant, cf. [16, 4]. Critical points of $P_{S^{k}}$ are called Paneitz-biharmonic maps in the following sense.

Definition 1.4**.**

A map $u\in C^{4}(S^{4},N)$ is called Paneitz-biharmonic if

[TABLE]

holds true for any $x\in S^{4}$. Analogously, a map $u\in C^{4}(S^{4}_{+},N)$ is called Paneitz-biharmonic with Dirichlet datum $g$ if $(u,Du)=(g,Dg)$ on $\partial S^{4}_{+}$ and (4) holds true for any $x\in S^{4}_{+}$.

Definition 1.5**.**

We say that the Riemannian manifold $N$ does not carry any non-constant Paneitz-biharmonic $4$-spheres if every Paneitz-biharmonic map $u\in C^{4}(S^{4},N)$ is constant.

Analogously, we say that $N$ does not carry any non-constant Paneitz-biharmonic $4$-halfspheres with constant boundary values if every Paneitz-biharmonic map $u\in C^{4}(S^{4}_{+},N)$ with $(u,Du)=(c,0)$ on $\partial S^{4}_{+}$ for some constant $c\in N$ satisfies $u=c$ on $S^{4}_{+}$.

1.2. Statement of the results

Now we are in a position to state our main results. In all statements, we restrict ourselves to the case of a flat boundary, i.e. to the case that $\Omega$ is a half ball $B_{R}^{+}$, with boundary values prescribed on the flat part of the boundary, which we denote by $T_{R}$. The general case of a smooth boundary can be reduced to this case by flattening the boundary. However, this procedure will change the Euclidean metric to a more general Riemannian one. Nevertheless, we decided to treat only the model case of the Euclidean metric in order not to overburden this work with additional technicalities.

Our first main result is the following compactness property for bounded sequences of variationally biharmonic maps.

Theorem 1.6**.**

Let $N\subset{\mathds{R}}^{L}$ be a compact, smooth Riemannian manifold that does neither carry non-constant Paneitz-biharmonic $4$-spheres nor non-constant Paneitz-biharmonic $4$-halfspheres with constant boundary values. Assume that $g_{i}\in C^{\infty}(B_{4}^{+},N)$, $i\in{\mathds{N}}$, is a sequence of boundary values and $u_{i}\in W^{2,2}(B_{4}^{+},N)$ are variationally biharmonic maps with respect to the Dirichlet data $g_{i}$ on $T_{4}$, and that both sequences are bounded in the sense

[TABLE]

for some $\alpha\in(0,1)$. Then there is a map $u\in W^{2,2}(B_{1}^{+},N)$ so that after passing to a subsequence, we have the convergence

[TABLE]

Remark 1.7**.**

The assumption on the non-existence of Paneitz-biharmonic $4$-spheres and $4$-halfspheres is necessary in the following sense. Assume that there is a non-constant Paneitz-biharmonic $4$-halfsphere with constant boundary values. Then, by means of stereographic projection and the conformal invariance of the Paneitz-bienergy, we infer a non-constant biharmonic map $u\in C^{\infty}({\mathds{R}}^{4}_{+},N)$ of finite bi-energy with $(u,Du)=(c,0)$ on $\partial{\mathds{R}}^{4}_{+}$. This map gives rise to the sequence of rescaled biharmonic maps

[TABLE]

for any sequence $\lambda_{i}\to\infty$, with the abbreviation $B_{r}^{4,+}$ for the four-dimensional upper halfball of radius $r>0$. For this sequence of biharmonic maps, we have

[TABLE]

which implies $\sup_{i\in{\mathds{N}}}\|u_{i}\|_{W^{2,2}(B_{1}^{4,+})}<\infty$ by standard $L^{2}$-estimates for the Laplace operator. Moreover, for an arbitrary map $\varphi\in C^{0}(\overline{B_{1}^{4,+}})$ with compact support, we compute

[TABLE]

which corresponds to the convergence of measures \mathcal{L}^{m}\hskip 1.00006pt\mbox{\LARGE\llcorner}\hskip 0.50003pt|\Delta v_{i}|^{2}\rightharpoondown c\delta_{0} in the limit $i\to\infty$, with the Dirac measure $\delta_{0}$ and the constant $c=\int_{{\mathds{R}}^{4}_{+}}|\Delta u|^{2}\mathrm{d}y>0$. The latter convergence contradicts the subconvergence of the sequence $u_{i}$ with respect to the $W^{2,2}$-norm.

Analogously, the existence of a non-constant Paneitz-biharmonic $4$-sphere yields a sequence of biharmonic maps $u_{i}$ on the full ball $B_{1}^{4}\subset{\mathds{R}}^{4}$ that does not contain a strongly convergent subsequence. This demonstrates the necessity of the assumptions on Paneitz-biharmonic spheres and halfspheres for our compactness result. **

The preceding compactness result is the crucial step for the derivation of the full boundary regularity for variationally biharmonic maps.

Theorem 1.8**.**

Let $N\subset{\mathds{R}}^{L}$ be a smooth compact Riemannian manifold that does neither carry a non-constant Paneitz-biharmonic $4$-sphere nor a non-constant Paneitz-biharmonic $4$-halfsphere with constant boundary values. Assume that the map $u\in W^{2,2}(B_{1}^{+},N)$ is variationally biharmonic with $(u,Du)=(g,Dg)$ on $T_{1}$ in the sense of traces, for Dirichlet values $g\in C^{\infty}(B_{1}^{+},N)$. Then $u$ is smooth in a full neighbourhood of $T_{1}$.

1.3. Plan of the paper

In Section 2, we gather some technical tools that will be crucial for our arguments. In particular, we derive a Morrey space estimate that is a consequence of a boundary monotonicity formula, and we recall some partial regularity results for variationally biharmonic maps. Moreover, we prove a gradient estimate under a smallness assumption on the energy.

Section 3 is then devoted to the proof of our compactness result. We introduce the notion of defect measure for a sequence of variationally biharmonic maps and analyse in what sense this measure detects the lack of strong convergence. We show that if a nontrivial defect measure exists, then a blow-up procedure yields a flat defect measure that is supported on an $(m-4)$-dimensional plane. This means that we can find a sequence of variationally biharmonic maps that converge strongly away from this plane. In this more controlled situation, it is then possible to follow the ideas by Lin [10] and to construct a suitable blow-up sequence around carefully chosen blow-up points. Depending on whether these points approach the boundary or not, we can show that the limit gives rise to a non-constant Paneitz-biharmonic $4$-halfsphere with constant boundary values or to a corresponding full $4$-sphere. Since the existence of such maps is excluded by assumption, we deduce the desired strong compactness for any bounded sequence of variationally biharmonic maps.

The next Section 4 contains some Liouville type results for biharmonic maps on half-spaces that arise as tangent maps of biharmonic maps in boundary points. Since we can show that all possible tangent maps are constant, the implementation of Federer’s dimension reduction principle in Section 5 allows us to deduce our second main result on the full boundary regularity of variationally biharmonic maps.

**Acknowledgments. **This work has been supported by the DFG-project SCHE 1949/1-1 “Randregularität biharmonischer Abbildungen zwischen Riemann’schen Mannigfaltigkeiten”.

2. Preliminaries

2.1. Notation

We write $B_{r}(a)\subset{\mathds{R}}^{m}$ for the open ball with radius $r>0$ and center $a\in{\mathds{R}}^{m}$ and $S_{r}(a):=\partial B_{r}(a)$ for the corresponding sphere. For the upper halfspace, we use the abbreviation ${\mathds{R}}^{m}_{+}:={\mathds{R}}^{m-1}\times[0,\infty)$, and write $B_{r}^{+}(a):=B_{r}(a)\cap{\mathds{R}}^{m}_{+}$ for arbitrary centers $a\in{\mathds{R}}^{m}_{+}$. Moreover, in the case of a center $a=(a^{\prime},0)\in\partial{\mathds{R}}^{m}_{+}$ we use the abbreviations $S_{r}^{+}(a):=S_{r}(a)\cap{\mathds{R}}^{m}_{+}$ for the curved part and $T_{r}(a):=B_{r}(a^{\prime})\times\{0\}$ for the flat part of the boundary of $B_{r}^{+}(a)$. If the center is clear from the context, we will often omit it in the notation and simply write $B_{r}$, $B_{r}^{+}$, $S_{r}$, $S_{r}^{+}$, $T_{r}$ instead of $B_{r}(a)$, $B_{r}^{+}(a)$, $S_{r}(a)$, $S_{r}^{+}(a)$, $T_{r}(a)$.

For the Lebesgue measure on ${\mathds{R}}^{m}$ we write $\mathcal{L}^{m}$, and for $0\leq k\leq m$, the $k$-dimensional Hausdorff measure on ${\mathds{R}}^{m}$ will be abbreviated by $\mathcal{H}^{k}$.

For $f\in L^{2}(B_{2}^{+})$ and $\lambda>0$, we write

[TABLE]

Finally, the singular set of a map $u:\Omega\to{\mathds{R}}^{L}$ is defined by

[TABLE]

2.2. Monotonicity formula and consequences

The proof of the following lemma can be retrieved from [1, Lemma 2.1].

Lemma 2.1**.**

Let $u\in W^{2,2}(\Omega,N)$ be a variationally biharmonic map with respect to the Dirichlet datum $g$ on $\Gamma\subset\partial\Omega$. Then for every $\xi\in C^{\infty}(\Omega\cup\Gamma,{\mathds{R}}^{m})$ with $\xi(x)\in T_{x}(\partial\Omega)$ for every $x\in\Gamma$ and $\operatorname{spt}\xi\Subset\Omega\cup\Gamma$, we have

[TABLE]

with the orthogonal projection $\Pi(u(x)):{\mathds{R}}^{L}\to T_{u(x)}N$.

The preceding lemma is the first step in the derivation of a boundary monotonicity formula, which has been proven in [1] for the case of a flat boundary. More precisely, we specialize to the case $\Omega=B_{R}^{+}(a)$ and $\Gamma=T_{R}(a)$.

Theorem 2.2**.**

([1, Theorem 1.2])* Let $m\geq 5$, $0<R\leq 1$ and assume that $u\in W^{2,2}(B_{R}^{+}(a),N)$ satisfies (5). Then, for a.e. radii $0<\rho<r<R$, we have the monotonicity formula*

[TABLE]

where we abbreviated

[TABLE]

with the short-hand notation $\partial_{X}:=(x_{i}-a_{i})\partial_{i}$, and

[TABLE]

In the above formula, the constants $\chi,K\geq 0$ depend only on the data $m,N$ and $\|Dg\|_{C^{2}}$. In particular, the constants $\chi$ and $K$ vanish in the limit $\|Dg\|_{C^{2}}\to 0$.

An important consequence of the preceding monotonicity formula is the following Morrey space estimate.

Lemma 2.3**.**

Let $m\geq 5$ and $g\in C^{3}(B_{1}^{+},N)$ be given. Assume that $u\in W^{2,2}(B_{1}^{+},N)$ satisfies (5) and that $(u,Du)=(g,Dg)$ holds on $T_{1}$, in the sense of traces. Then for any ball $B_{R}^{+}(a)\subset B_{1}^{+}$ with radius $R\in(0,1]$ we have

[TABLE]

with constants $c_{1}$, $c_{2}$ that depend at most on $m,N,$ and $\|Dg\|_{C^{2}}$. Moreover, we have $c_{2}\to 0$ in the limit $\|Dg\|_{C^{2}}\to 0$.

Proof.

For the proof, we modify some ideas from [27] and [15]. Throughout this proof, we write $c$ for a constant that may depend on $m$, $N$, and $\|Dg\|_{C^{2}}$. We first consider the case $a\in T_{1}$. For $0<s<\frac{R}{8}$, we consider radii $\rho\in[2s,4s]$ and $r\in[\frac{R}{2},R]$ that will be chosen later. Our goal is to derive an estimate for the term

[TABLE]

where $\chi\geq 0$ is the constant from Theorem 2.2, which depends only on $m,N,$ and $\|Dg\|_{C^{2}}$. For the difference of this term and the term $\Phi_{u}(a;\rho)$ defined in (7), we compute, using again the abbreviation $X(x):=x-a$,

[TABLE]

where we applied Young’s inequality in the last step. Taking the mean integral over $\rho\in[2s,4s]$, we deduce

[TABLE]

where we used $\frac{\rho}{2}\leq 2s$ and $4s\leq\frac{R}{2}\leq r$ in the last step. Both terms on the right-hand side can be estimated by an application of the monotonicity formula (6). This leads to the estimate

[TABLE]

for a.e. $r\in[\frac{R}{2},R]$, with the constant $K=K(m,N,\|Dg\|_{C^{2}})$ from Theorem 2.2. We recall that $K\to 0$ as $\|Dg\|_{C^{2}}\to 0$. By definition of $\Psi_{u}$, the last integral can be estimated by

[TABLE]

Integrating by parts twice, we estimate the second last integral as follows.

[TABLE]

where we introduced the notation

[TABLE]

Combining the three preceding estimates and recalling the definition of $\Phi_{u}$, we deduce

[TABLE]

for a.e. $r\in[\frac{R}{2},R]$. Now we choose a good radius $r\in[\frac{R}{2},R]$ in the sense that

[TABLE]

Moreover, this radius can be chosen in such a way that the preceding estimate is valid for this choice of $r$, which implies

[TABLE]

Our next aim is to estimate $\widetilde{\Phi}_{u}(a;\rho)$ from below. First, we observe that with a standard cut-off function $\eta\in C^{\infty}_{0}(B_{\rho}(a))$ with $\eta\equiv 1$ in $B_{\rho/2}(a)$, two integrations by parts and Young’s inequality lead to the estimate

[TABLE]

which implies

[TABLE]

where $c=c(m)$. Two applications of Gauß’ theorem yield

[TABLE]

Since $m>3$, we can re-absorb the integral of $|Du|^{2}$ into the left-hand side. Multiplying the resulting estimate by $\rho^{2-m}$, we infer

[TABLE]

Combining estimates (11) and (12), we deduce

[TABLE]

for any $\rho\in[2s,4s]$, which implies in turn

[TABLE]

We use this to estimate the left-hand side of (10) from below, with the result

[TABLE]

for every $s\in(0,\frac{R}{8})$, where we abbreviated $\widetilde{K}:=K+\|Dg\|_{C^{1}}^{2}$. We take the supremum over $s\in[\delta,\frac{R}{8})$ on both sides, for some $\delta>0$, and infer

[TABLE]

Now we choose a radius $R_{0}=R_{0}(m,N,\|Dg\|_{C^{2}})>0$ so small that $cR_{0}\leq\frac{1}{2}$, which allows us to re-absorb the last term into the left-hand side, provided $R\leq R_{0}$. Letting $\delta\downarrow 0$, we deduce

[TABLE]

for all $s\leq R\leq R_{0}$, in the case $a\in T_{1}$. For radii $s,R$ with $s\leq R_{0}\leq R\leq 1$, we obtain the same estimate by applying (13) with $R_{0}$ in place of $R$ and then enlarging the domain of integration on the right-hand side. Finally, the estimate (13) is immediate in the case $R_{0}\leq s\leq R$, since $R_{0}$ is a universal constant. Hence, we obtain (13) for any $s\leq R\leq 1$ and boundary points $a\in T_{1}$.

In the interior case $B_{R}(a)\subset B^{+}$, we can argue in the same way, starting from the interior version of the monotonicity formula from [2], to derive the estimate

[TABLE]

for all $s\leq R$. The two preceding estimates can be combined in a standard way to obtain the result. In fact, let $B_{\rho}^{+}(y)\subset B_{R/2}^{+}(a)$ be an arbitrary ball with $\rho\leq\frac{R}{4}$ and $y_{m}\leq\frac{R}{4}$. We use the notation $(y^{\prime},y_{m}):=y\in{\mathds{R}}^{m-1}\times{\mathds{R}}$ and let $R_{1}:=\max\{y_{m},\rho\}$. Then we use first the interior estimate (14) and then the boundary version (13) to deduce

[TABLE]

In the remaining case $y_{m}>\frac{R}{4}$, the corresponding result follows from the interior estimate (14). Finally, for radii $\rho>\frac{R}{4}$, the above estimate is trivial. We note that $\widetilde{K}\to 0$ in the limit $\|Dg\|_{C^{2}}\to 0$. Hence, we have established the assertion in any case. ∎

For later reference, we state another consequence of the monotonicity formula.

Corollary 2.4**.**

Assume that $u\in W^{2,2}(B_{1}^{+},N)$ satisfies (5) and $(u,Du)=(g,Dg)$ on $T_{1}$, in the sense of traces. Then the limit

[TABLE]

exists for any $a\in T_{1}$.

Proof.

For two radii $0<\rho<\frac{R}{2}<\frac{1}{4}(1-|a|)$, the monotonicity formula from Theorem 2.2 implies

[TABLE]

Using an integration by parts argument similarly to (9) and then applying Lemma 2.3, we can estimate

[TABLE]

Here, $c(u)$ denotes a constant that depends on $m,N,\|Dg\|_{C^{2}}$, and $\|u\|_{W^{2,2}}$. We use this to estimate the right-hand side of (15). Then, we first let $\rho\downarrow 0$ and then $R\downarrow 0$ in the resulting estimate, which implies

[TABLE]

This concludes the proof of the corollary. ∎

2.3. Partial regularity for variationally biharmonic maps

The following $\varepsilon$-regularity result was first established in [7, Thm. 1.1] under the additional assumption that a boundary monotonicity inequality of the type (6) is satisfied. The proof was later completed in [1], where the boundary monotonicity formula was proved for arbitrary variationally biharmonic maps. The Morrey space estimate that results from the monotonicity formula is stated in Lemma 2.3. Combining [7, Lemma 3.1] with Lemma 2.3 leads to the following regularity result.

Theorem 2.5**.**

Let $m\geq 5$, $a\in\partial{\mathds{R}}^{m}_{+}$, $\rho>0$, and $g\in C^{\infty}(B_{\rho}^{+}(a),N)$ be given. There exists a constant $\varepsilon_{1}>0$, depending only on $m,N$, and $\|Dg\|_{C^{2}}>0$ so that for every weakly biharmonic map $u\in W^{2,2}(B_{\rho}^{+}(a),N)$ with (5) that attains the Dirichlet datum $g$ on $T_{\rho}(a)$ and fulfills the estimate

[TABLE]

we have $u\in C^{\infty}(B_{\rho/2}^{+}(a),N)$.

In a standard way, the preceding theorem and its interior counterpart from [30] imply the following partial regularity result.

Corollary 2.6**.**

Let $u\in W^{2,2}(B_{4}^{+},N)$ be variationally biharmonic with respect to the Dirichlet datum $g\in C^{\infty}(B_{4}^{+},N)$ on $T_{4}$. Then there is a subset $\Sigma\subset\Omega$ with $\mathcal{H}^{m-4}(\Sigma)=0$ so that $u\in C^{\infty}(\Omega\setminus\Sigma,N)$.

Finally, we have the following quantitative estimate. In the case of harmonic maps, the corresponding result is due to Schoen [23]. For the higher order case considered here, we employ a technique that goes back to Moser [14, Lemma 5.3], see also [22, Lemma 5.3].

Lemma 2.7**.**

For every $\delta>0$, $\Lambda>0$, and $\alpha\in(0,1)$, there is a constant $\varepsilon=\varepsilon(\delta,\Lambda,\alpha,m,N)>0$ so that the following holds. Assume that $u\in C^{4}(\overline{B}_{1}^{+},N)$ is a biharmonic map with Dirichlet values $g\in C^{4,\alpha}(B_{1}^{+},N)$ on $T_{1}$ for which

[TABLE]

holds true, then we have the estimate

[TABLE]

Proof.

We use the abbreviation $[u]_{C^{4}}(x):=\sum_{k=1}^{4}|D^{k}u(x)|^{1/k}$ for $x\in B_{1}^{+}$. If the assertion of the lemma was not true, we could find sequences of biharmonic maps $u_{i}\in C^{4}(\overline{B}_{1}^{+},N)$ and boundary values $g_{i}\in C^{4,\alpha}(B_{1}^{+},N)$ with $(u_{i},Du_{i})=(g_{i},Dg_{i})$ on $T_{1}$ for all $i\in{\mathds{N}}$ so that

[TABLE]

but for all $i\in{\mathds{N}}$, we have

[TABLE]

For every $i\in{\mathds{N}}$, we choose a radius $r_{i}\in[0,1)$ with

[TABLE]

and then a point $x_{i}\in\overline{B}_{r_{i}}^{+}$ with

[TABLE]

With these choices, we define scaling factors by

[TABLE]

We observe that (20) implies $\lambda_{i}<\frac{1-r_{i}}{2}<1$. With these factors we define rescaled maps

[TABLE]

for $x\in\Omega_{i}$, where

[TABLE]

The rescaled maps satisfy

[TABLE]

by definition of $\lambda_{i}$. Moreover, because of $B_{\lambda_{i}}(x_{i})\subset B_{(1+r_{i})/2}$ and the choice of $r_{i}$, $x_{i}$, and $\lambda_{i}$, we infer

[TABLE]

From $\lambda_{i}<1$ and (19)2, we obtain

[TABLE]

By the scaling invariance of the biharmonic map equation, the maps $v_{i}$ are again biharmonic, which means by (1) that they satisfy a boundary value problem of the form

[TABLE]

Clearly, in the case that the last set is empty, the map $v_{i}$ satisfies the differential equation on the full ball $B_{1}$ and there is no boundary condition. From the form of the biharmonic map equation and (22), we infer

[TABLE]

For a standard cut-off function $\zeta\in C^{\infty}_{0}(B_{1},[0,1])$ with $\zeta\equiv 1$ in $B_{1/2}$, we use classical Schauder estimates for the maps $\zeta v_{i}$ on the halfspaces ${\mathds{R}}^{m}\cap\{x^{(m)}=-\lambda_{i}^{-1}x_{i}^{(m)}\}$. In this way, we deduce that (24) implies

[TABLE]

In view of (22), (23), and (25), we infer that the restrictions $v_{i}|_{B_{1/2}^{+}}$ are bounded in $C^{4,\alpha}(B_{1/2}^{+},N)$, independently of $i\in{\mathds{N}}$. Therefore, after passing to a subsequence, the Arzelà-Ascoli theorem yields the convergence

[TABLE]

for some limit map $v\in C^{4}(B_{1/2}^{+},N)$. Now on the one hand, the identity (21) implies

[TABLE]

but on the other hand, the choice of $u_{i}$ according to (19)1 leads to the estimate

[TABLE]

which means that $v$ is constant on $B_{1/2}^{+}$. In view of (26), this yields the desired contradiction and completes the proof of the lemma. ∎

By combining the preceding regularity results, we arrive at the following conclusion.

Corollary 2.8**.**

We consider a ball $B_{r}^{+}(x_{0})=B_{r}(x_{0})\cap{\mathds{R}}^{m}_{+}$ for some $x_{0}\in{\mathds{R}}^{m}_{+}$ and $r\in(0,1]$. There is a constant $\varepsilon_{0}=\varepsilon_{0}(\Lambda,\alpha,m,N)>0$, so that for every weakly biharmonic map $u\in W^{2,2}(B_{r}^{+}(x_{0}),N)$ that satisfies (5) and attains the Dirichlet datum $g\in C^{\infty}(B_{r}^{+}(x_{0}),N)$ on $T_{r}(x_{0})$, the estimates

[TABLE]

imply $u\in C^{4}(B_{r/2}^{+}(x_{0}),N)$ with $\|Du\|_{C^{3}(B_{r/2}^{+}(x_{0}))}\leq c(\Lambda,\alpha,m,N)$.

Proof.

By a scaling argument, it suffices to consider the case $r=1$. In the case that $B_{3/4}^{+}(x_{0})$ intersects $\partial{\mathds{R}}^{m}_{+}$, we apply Lemma 2.3 on balls $B_{1/4}^{+}(a)$ for any $a\in B_{3/4}^{+}(x_{0})\cap\partial{\mathds{R}}^{m}_{+}$. In view of this lemma and assumption (27)1, we infer

[TABLE]

for any $R_{0}\in(0,\tfrac{1}{8})$. By choosing first $R_{0}$ and then $\varepsilon_{0}$ small enough in dependence on $\Lambda,\alpha,m$, and $N$, we can ensure that the assumptions of Theorem 2.5 and Lemma 2.7 with $\delta=1$ are satisfied on the ball $B_{R_{0}}^{+}(a)$, which imply that $u\in C^{4}(B_{R_{0}/2}^{+}(a))$ with $\|Du\|_{C^{3}(B_{R_{0}/4}^{+}(a))}\leq c$. We note that the application of Lemma 2.7 is possible after a suitable rescaling. Since the last estimate holds for any $a\in B_{3/4}^{+}(x_{0})\cap\partial{\mathds{R}}^{m}_{+}$, we deduce the asserted gradient estimates in every point $y\in B_{1/2}^{+}(x_{0})$ with $y_{m}<\frac{1}{4}R_{0}$. In the remaining case $y\in B_{1/2}^{+}(x_{0})$ with $y_{m}\geq\tfrac{1}{4}R_{0}$, we apply the corresponding interior estimates on the ball $B_{R_{0}/4}(y)$, cf. [21, Thm. 2.6]. In this way, we arrive at the desired bound $\|Du\|_{C^{3}(B_{1/2}^{+}(x_{0}))}\leq c(\Lambda,\alpha,m,N)$. ∎

3. Compactness for sequences of variationally biharmonic maps

3.1. The defect measure

Our goal is to prove compactness for a sequence of variationally biharmonic maps $u_{i}\in W^{2,2}(B_{4}^{+},N)$ with respect to Dirichlet values $g_{i}\in C^{4,\alpha}(B_{4}^{+},N)$ on $T_{4}$. We assume that the sequence is bounded in the sense that

[TABLE]

More precisely, we consider the slightly more general case of maps with (1) and (5) instead of variationally biharmonic maps, since the properties (1) and (5) are clearly preserved under strong convergence in $W^{2,2}$. In view of (28), Lemma 2.3 implies the Morrey space bound

[TABLE]

for a constant $\Lambda\geq 1$ that depends on $m,N$, and the sequences $(u_{i})$ and $(g_{i})$. Here we used the Morrey type norms that are defined by

[TABLE]

for $f\in L^{2}(B_{2}^{+})$. From now on, we restrict ourselves to the ball $B_{2}^{+}$ and assume that a bound of the type (29) is valid. This bound implies in particular that the sequence of Radon measures \mathcal{L}^{m}\hskip 1.00006pt\mbox{\LARGE\llcorner}\hskip 0.50003pt|\Delta u_{i}|^{2} is bounded on $\overline{B}_{2}^{+}$. Therefore, possibly after passing to a subsequence, we infer a map $u\in W^{2,2}(B_{2}^{+},N)$ and a Radon measure $\nu$ on $\overline{B}_{2}^{+}$ with $u_{i}\rightharpoondown u$ weakly in $W^{2,2}(B_{2}^{+},{\mathds{R}}^{L})$, strongly in $W^{1,2}(B_{2}^{+},{\mathds{R}}^{L})$ and a.e., and moreover

[TABLE]

weakly* in the space of Radon measures, as $i\to\infty$. The lower semicontinuity of the $L^{2}$-norm with respect to weak convergence implies $\nu\geq 0$. We call the measure $\nu$ the defect measure of the sequence $u_{i}$, since it detects a possible lack of strong convergence in $W^{2,2}$, see Lemma 3.2 below. The pair $(u,\nu)$ can be considered as the limit configuration of the sequence $u_{i}$. This motivates the following

Definition 3.1**.**

For sequences of maps $u_{i}\in W^{2,2}(B_{2}^{+},N)$ and nonnegative Radon measures $\nu_{i}$ on $\overline{B}_{2}^{+}$, where $i\in{\mathds{N}}_{0}$, we write $(u_{i},\nu_{i})\rightrightarrows(u_{0},\nu_{0})$ as $i\to\infty$ if and only if convergence holds in the following sense.

[TABLE]

For the set of all limit configurations of biharmonic maps, we write

[TABLE]

for a given constant $\Lambda\geq 1$ and $\alpha\in(0,1)$. Here, $"0"$ denotes the zero measure and we used the Morrey norms defined in (30). For a given pair $\mu=(u,\nu)\in\mathcal{B}_{\Lambda}$, we define the energy concentration set $\Sigma_{\mu}$ as the set of points $a\in\overline{B}_{2}^{+}$ with the property

[TABLE]

where the constant $\varepsilon_{0}=\varepsilon_{0}(\Lambda,\alpha,m,N)>0$ is chosen according to Corollary 2.8.

The following lemma clarifies the meaning of the defect measure and the energy concentration set.

Lemma 3.2**.**

Assume that $u_{i}\in W^{2,2}(B_{2}^{+},N)$, $i\in{\mathds{N}}$, is a sequence of maps with (1), (5), boundary values

[TABLE]

so that the bound

[TABLE]

is satisfied. Moreover, we assume that $(u_{i},0)\rightrightarrows(u,\nu)=:\mu$ as $i\to\infty$, for some $(u,\nu)\in\mathcal{B}_{\Lambda}$. Then there holds

1. (i)

$u_{i}\to u$* in $C^{3}_{\textup{loc}}(B_{2}^{+}{\setminus}\Sigma_{\mu},{\mathds{R}}^{L})$ as $i\to\infty$.* 2. (ii)

If the defect measure satisfies $\operatorname{spt}\nu\cap\overline{B}_{1}^{+}=\varnothing$, then we have strong convergence $u_{i}\to u$ in $W^{2,2}(B_{1}^{+},{\mathds{R}}^{L})$, as $i\to\infty$.

Proof.

In order to prove (i), we choose an arbitrary point $a\in B_{2}^{+}{\setminus}\Sigma_{\mu}$. By the definition of $\Sigma_{\mu}$, we may choose a $\rho\in(0,1)$ with

[TABLE]

By slightly diminishing the value of $\rho$ if necessary, we can additionally achieve that $\nu(\partial B_{\rho}^{+}(a))=0$, because $\nu(\partial B_{\rho}^{+}(a))>0$ can hold at most for countably many values of $\rho\in(0,1)$. Using the convergence $(u_{i},0)\rightrightarrows(u,\nu)$, we conclude

[TABLE]

Therefore, Corollary 2.8 yields the uniform estimate $\|u_{i}\|_{C^{4}(B_{\rho/2}^{+}(a))}\leq c(\Lambda,\alpha,m,N)$ for all sufficiently large $i\in{\mathds{N}}$, from which we infer by Arzéla-Ascoli’s theorem that $u_{i}\to u$ holds in $C^{3}(B_{\rho/2}^{+}(a),N)$ as $i\to\infty$. Since $a\in B_{2}^{+}{\setminus}\Sigma_{\mu}$ was arbitrary, this implies (i).

For the proof of (ii), we note that in the case of $\operatorname{spt}\nu\cap\overline{B}_{1}^{+}=\varnothing$, the set $\Sigma_{\mu}\cap\overline{B}_{1}^{+}$ is given by

[TABLE]

Since the values of $u$ are contained in the bounded manifold $N$, we have $u\in L^{\infty}\cap W^{2,2}(B_{1}^{+},N)$, which implies by Gagliardo-Nirenberg embedding that $u\in W^{1,4}(B_{1}^{+},N)$. Therefore, we infer from Hölder’s inequality and [31, Lemma 3.2.2] that $\mathcal{H}^{m-4}(\Sigma_{\mu}\cap\overline{B}_{1}^{+})=0$. This implies that for any given $\varepsilon>0$, we can choose a cover $A:=\cup_{k\in{\mathds{N}}}B_{\rho_{k}}(a_{k})\supset\Sigma_{\mu}\cap\overline{B}_{1}^{+}$ of open balls with radii $\rho_{k}\in(0,1)$ and centers $a_{k}\in\overline{B}_{1}^{+}$ so that $\sum_{k\in{\mathds{N}}}\rho_{k}^{m-4}<\varepsilon$. Hence, the bound (31) yields the following estimate for all $i\in{\mathds{N}}$.

[TABLE]

On the compact set $\overline{B}_{1}^{+}\setminus A\subset\overline{B}_{1}^{+}{\setminus}\Sigma_{\mu}$, the conclusion (i) implies $u_{i}\to u$ in $C^{3}(\overline{B}_{1}^{+}\setminus A,{\mathds{R}}^{L})$. Hence,

[TABLE]

Since $\varepsilon>0$ was arbitrary, we conclude that $u_{i}\to u$ holds in $W^{2,2}(B_{1}^{+},{\mathds{R}}^{L})$ in the limit $i\to\infty$, as claimed. ∎

Next, we analyse the relation between the defect measure and the energy concentration set.

Lemma 3.3**.**

There are positive constants $c_{1},c_{2}$, depending only on $m$, so that every pair $\mu=(u,\nu)\in\mathcal{B}_{\Lambda}$ satisfies

[TABLE]

Furthermore, $\Sigma_{\mu}$ is a closed set and $\Sigma_{\mu}=\operatorname{sing}(u)\cup\operatorname{spt}(\nu)$.

Proof.

The inclusion $\operatorname{sing}(u)\cup\operatorname{spt}(\nu)\subset\Sigma_{\mu}$ holds by Corollary 2.8 and Lemma 3.2 (i). For the converse inclusion, we assume that there is some point $a\in\Sigma_{\mu}\setminus\operatorname{sing}(u)$. By the choice of $a$, the functions $|\Delta u|$ and $|Du|$ are bounded on a neighborhood of $a$. Therefore, the definition of $\Sigma_{\mu}$ implies

[TABLE]

from which we infer $a\in\operatorname{spt}(\nu)$. We have thus proven $\Sigma_{\mu}=\operatorname{sing}(u)\cup\operatorname{spt}(\nu)$, which implies in particular that $\Sigma_{\mu}$ is a closed set.

Now we turn our attention to the proof of (32). For a Borel set $A\subset\Sigma_{\mu}\cap\overline{B}_{1}^{+}$, we choose an arbitrary cover $\cup_{j\in{\mathds{N}}}B_{\rho_{j}}(a_{j})\supset A$ of balls with radii $\rho_{j}\in(0,1)$ and centers $a_{j}\in\overline{B}_{1}^{+}$. Since $(u,\nu)\in\mathcal{B}_{\Lambda}$, we infer that

[TABLE]

holds true, for a constant $c_{2}=c_{2}(m)$. Since the cover of $A$ was arbitrary, we conclude $\nu(A)\leq c_{2}\Lambda\mathcal{H}^{m-4}(A)$. For the proof of the first estimate in (32), we choose a set $E\subset\Sigma_{\mu}$ with $\mathcal{H}^{m-4}(E)=0$ and

[TABLE]

A set with this property exists by [31, Lemma 3.2.2]. From the choice of $E$ and the definition of $\Sigma_{\mu}$, we know

[TABLE]

Let $\varepsilon>0$ be given. By the definition of the Hausdorff measure, we may choose $\delta>0$ small enough to ensure

[TABLE]

where $\alpha(m-4)$ denotes the volume of the $(m-4)$-dimensional unit ball. Since $\nu$ is a Radon measure, we may choose an open set $O_{\varepsilon}\supset A$ with $\nu(O_{\varepsilon}\cap{\mathds{R}}^{m}_{+})\leq\nu(A)+\varepsilon$. We consider the family of balls $B_{\rho_{j}}(a_{j})\subset O_{\varepsilon}$ with $0<\rho_{j}\leq\delta/5$, centers $a_{j}\in A\setminus E$ and the property

[TABLE]

By (33), the union of all balls with these properties covers the set $A\setminus E$. A Vitali type covering argument therefore yields the existence of a countable disjoint family $\{B_{\rho_{j}}(a_{j})\}_{j\in{\mathds{N}}}$ of balls with the property (34), $B_{\rho_{j}}(a_{j})\subset O_{\varepsilon}$ and $A\setminus E\subset\,\cup_{j}B_{5\rho_{j}}(a_{j})$, where $5\rho_{j}\leq\delta$ for all $j\in{\mathds{N}}$. This implies

[TABLE]

by the choice of $\delta$. Since the balls $B_{\rho_{j}}(a_{j})$ are pairwise disjoint and satisfy (34), we can further estimate

[TABLE]

Combining the last two estimates, we arrive at

[TABLE]

with a constant $c_{1}=c_{1}(m)>0$. Since $\varepsilon>0$ was arbitrary, we can omit the terms involving $\varepsilon$ in the preceding estimate. This completes the proof of the lemma. ∎

For the blow-up analysis of the defect measure, the case of a constant limit map is of particular interest. In this case, the defect measure inherits a monotonicity property from the sequence of biharmonic maps.

Lemma 3.4**.**

We consider a pair $(c,\nu)\in\mathcal{B}_{\Lambda}$, where $c\in N$ denotes a constant map. More precisely, we assume that $(c,\nu)$ can be approximated by biharmonic maps $u_{i}$ in the sense $\mathcal{B}_{\Lambda}\ni(u_{i},0)\rightrightarrows(c,\nu)$, where the approximating maps satisfy the boundary condition $(u_{i},Du_{i})=(g_{i},Dg_{i})$ on $T_{2}$, for boundary values $g_{i}\in C^{\infty}(B_{2}^{+},N)$ with

[TABLE]

Then, the functions

[TABLE]

are monotonically nondecreasing for every $a\in\overline{T}_{1}$. Moreover, there is a subsequence $\{i_{j}\}\subset{\mathds{N}}$ so that

[TABLE]

for a.e. $\rho,r\in(0,1]$, as $j\to\infty$, where the terms $\Phi_{u_{i}}$, $\Psi_{u_{i}}$ and $K_{i}$ are defined in Theorem 2.2.

Proof.

From Theorem 2.2 we deduce that the approximating biharmonic maps $u_{i}$ satisfy the monotonicity formula (6) with constants $\chi_{i}$, $K_{i}>0$ that satisfy $\chi_{i}\to 0$ and $K_{i}\to 0$ as a consequence of $\|Dg_{i}\|_{C^{3}(B_{2}^{+})}\to 0$ in the limit $i\to\infty$. Discarding the non-negative integral on the left-hand side of the monotonicity formula, we deduce

[TABLE]

for all $i\in{\mathds{N}}$ and a.e. radii $0<\rho<r<1$. Next, for a.e. radius $r\in(0,1)$ and $X(x):=x-a$, we may define

[TABLE]

as well as

[TABLE]

Since $u_{i}\to c$ strongly in $W^{1,2}(B_{2}^{+},{\mathds{R}}^{K})$, $\sup_{i}\|D^{2}u_{i}\|_{L^{2}(B_{2}^{+})}^{2}<\infty$, and $K_{i}\to 0$, we have $f_{i}\to 0$ and $g_{i}\to 0$ in $L^{1}([0,1])$, as $i\to\infty$. Therefore, there is a subsequence $\{i_{j}\}\subset{\mathds{N}}$ with $f_{i_{j}}\to 0$ a.e. and $g_{i_{j}}\to 0$ a.e. in $[0,1]$, as $j\to\infty$. We conclude that for almost every $r\in[0,1]$,

[TABLE]

in the limit $j\to\infty$, if we choose in particular $r\in[0,1]$ in such a way that $\nu(\partial B_{r}(a))=0$. Similarly, we deduce

[TABLE]

as $j\to\infty$, for a.e. $0<\rho<r<1$. The two preceding convergences imply assertion (36). Using this result in (37), we infer

[TABLE]

for a.e. $0<\rho<r\leq 1$. Since the measures $\nu(B_{\rho}^{+}(a)),$ $\nu(B_{r}^{+}(a))$ depend left-continuously on $\rho,r\in[0,1]$, this implies the asserted monontonicity property. ∎

3.2. Existence of flat tangent measures

We consider tangent pairs of $\mu=(u,\nu)\in\mathcal{B}_{\Lambda}$ in a boundary point $a\in\overline{T}_{1}$ in the following sense. For scaling factors $r\in(0,1)$ we define rescaled versions $\mu_{a,r}=(u_{a,r},\nu_{a,r})$ by

[TABLE]

We note that the definition of $\mathcal{B}_{\Lambda}$ implies

[TABLE]

for any $a\in\overline{T}_{1}$ and $r\in(0,\tfrac{1}{2})$. For a map $u_{*}\in W^{2,2}(B_{2}^{+},N)$ and a Radon measure $\nu_{*}$ on $\overline{B}_{2}^{+}$, we call $\mu_{*}=(u_{*},\nu_{*})$ a tangent pair of $\mu$ in $a\in\overline{T}_{1}$ if there exists a sequence $r_{i}\searrow 0$ so that $\mu_{a,r_{i}}\rightrightarrows\mu_{*}$. For the family of all tangent pairs of a given $\mu\in\mathcal{B}_{\Lambda}$ in a boundary point, we write

[TABLE]

A standard diagonal sequence argument yields

[TABLE]

cf. [21, Lemma 3.3]. More precisely, assume that $\mu_{a,r_{i}}\rightrightarrows\mu_{\ast}$ as $i\to\infty$, and let $u_{k}\in W^{2,2}(B_{2}^{+},N)$ be biharmonic maps as in Definition 3.1 with $(u_{k},0)\rightrightarrows\mu$ in the limit $k\to\infty$. Then a diagonal sequence argument yields a sequence $(k_{i})_{i\in{\mathds{N}}}$ in ${\mathds{N}}$ with $(\tilde{u}_{i},0):=((u_{k_{i}})_{a,r_{i}},0)\rightrightarrows\mu_{\ast}$ as $i\to\infty$, which proves $\mu_{\ast}\in\mathcal{B}_{\Lambda}$, and thereby the claim (40). We note that the boundary values $\tilde{g}_{i}:=(g_{k_{i}})_{a,r_{i}}\in C^{\infty}(B_{2}^{+},N)$ of $\tilde{u}_{i}$ satisfy

[TABLE]

This means that every tangent pair $\mu_{\ast}\in\mbox{T}_{b}(\mu)$ is the limit of biharmonic maps $\tilde{u}_{i}$ in the sense $(\tilde{u}_{i},0)\rightrightarrows\mu_{\ast}$, where the boundary values of $\tilde{u}_{i}$ converge to a constant $c_{\ast}\in N$ in the sense

[TABLE]

In the following lemma, we construct a flat defect measure by a double blow-up procedure.

Lemma 3.5**.**

Assume that $N$ does not carry a non-constant Paneitz-biharmonic $4$-sphere, and that there is a pair $\mu=(u,\nu)\in\mathcal{B}_{\Lambda}$ with $\operatorname{spt}\nu\cap\overline{B}_{1}^{+}\neq\varnothing$. Then there exists a pair $(c_{\ast},\bar{\nu})\in\mbox{T}_{b}(\mu)\subset\mathcal{B}_{\Lambda}$, where $c_{\ast}\in N$ represents a constant map and

[TABLE]

for an $(m-4)$-dimensional subspace $V\subset\partial{\mathds{R}}^{m}_{+}$ and a constant $c_{0}>0$.

Proof.

Since $u\in W^{2,2}\cap W^{1,4}(\overline{B}_{2}^{+},N)$, we know that for $\mathcal{H}^{m-4}$-a.e. $a\in\overline{T}_{1}$,

[TABLE]

cf. [31, Lemma 3.2.2]. Since $N$ does not carry a non-constant Paneitz-biharmonic $4$-sphere, we know from the analysis of the interior case in [21, Thm. 1.6] that $\operatorname{spt}\nu\subset\partial{\mathds{R}}^{n}_{+}$. On the other hand, we have $\operatorname{spt}\nu\cap\overline{B}_{1}^{+}\neq\varnothing$ by assumption, so that Lemma 3.3 implies $\mathcal{H}^{m-4}(\Sigma_{\mu}\cap\overline{T}_{1})>0$. Therefore, we may choose a point $a\in\Sigma_{\mu}\cap\overline{T}_{1}$ with the property (43). As a consequence of (43), there is a sequence $r_{i}\searrow 0$ so that the rescaled maps $u_{a,r_{i}}$ satisfy

[TABLE]

as $i\to\infty$, for some constant $c_{\ast}\in N$. In view of (39), by passing to a subsequence we can achieve the convergence $\mu_{a,r_{i}}\rightrightarrows(c_{\ast},\nu_{*})$ for some Radon measure $\nu_{*}$ on $\overline{B}_{2}^{+}$ as $i\to\infty$. The result of this first blow-up is a tangent pair $\mu_{*}:=(c_{\ast},\nu_{*})\in\mbox{T}_{b}(\mu)\subset\mathcal{B}_{\Lambda}$ for which the terms $r^{4-m}\,\nu_{*}(B_{r}^{+}(a))$ depend monotonically nondecreasing on $r\in(0,1)$ for every $a\in\overline{T}_{1}$, see Lemma 3.4. We point out that assumption (35) of this lemma is satisfied in view of (41). In particular, this monotonicity property implies that the $(m-4)$-dimensional density

[TABLE]

exists for every $a\in\overline{T}_{1}$. Because of $(c_{\ast},\nu_{*})\in\mathcal{B}_{\Lambda}$ we have $\Theta^{m-4}(\nu_{*},a)\leq\Lambda$ for every $a\in\overline{T}_{1}$. As already noted above, the interior result [21, Thm. 1.6] implies $\Sigma_{\mu_{\ast}}\subset\partial{\mathds{R}}^{m}_{+}$. Since $\mu_{*}=(c_{\ast},\nu_{*})$ with a constant map $c_{\ast}$, we can characterize the set $\Sigma_{\mu_{*}}\cap\overline{B}_{1}^{+}$ by

[TABLE]

Because Lemma 3.3 implies $\nu_{*}(\overline{B}_{1}^{+}\setminus\Sigma_{\mu_{*}})=0$, we deduce

[TABLE]

The above property implies the existence of an $(m-4)$-flat tangent measure of $\nu_{*}$, see [12, Thm. 14.18]. More precisely, for $\nu_{*}$-a.e. $y\in\overline{B}_{1}^{+}$, there is a sequence $\rho_{j}\searrow 0$ with (\nu_{*})_{y,\rho_{j}}\overset{\raisebox{-4.30554pt}{\scriptsize*}}{\rightharpoondown}\bar{\nu} weakly∗ in the space of Radon measures, where \bar{\nu}=c_{0}\mathcal{H}^{m-4}\hskip 1.00006pt\mbox{\LARGE\llcorner}\hskip 0.50003ptV for an $(m-4)$-dimensional subspace $V\subset{\mathds{R}}^{m}$ and a constant $c_{0}>0$. As above, we infer from the interior case in [21, Thm. 1.6] that $\nu_{*}(\overline{B}_{1}^{+}\setminus\overline{T}_{1})=0$, so that we can find a boundary point $y\in\overline{T}_{1}$ with the above property and with $V\subset\partial{\mathds{R}}^{n}_{+}$. Finally, we note that a diagonal sequence argument implies that tangent pairs of tangent pairs are again tangent pairs, which implies that $(c_{\ast},\bar{\nu})\in\mbox{T}_{b}(\mu)$. This completes the proof. ∎

3.3. Construction of a biharmonic $4$-sphere or $4$-halfsphere

Theorem 3.6**.**

Assume that there is a pair $\mu=(u,\nu)\in\mathcal{B}_{\Lambda}$ with $\operatorname{spt}\nu\cap\overline{B}_{1}^{+}\neq\varnothing$. Then there exists a non-constant Paneitz-biharmonic sphere $v\in C^{\infty}(S^{4},N)$ or a non-constant Paneitz-biharmonic half-sphere $C^{\infty}(S_{+}^{4},N)$ with constant boundary values.

Proof.

For the proof we will adapt techniques from [10], see also [21] for the higher order case. Assume for contradiction that $N$ does not contain any non-constant Paneitz-biharmonic $4$-spheres. Then, after a suitable rotation, Lemma 3.5 yields the existence of a pair $\bar{\mu}=(c_{\ast},\bar{\nu})\in\mbox{T}_{b}(\mu)\subset\mathcal{B}_{\Lambda}$, where $c_{\ast}\in N$ is a constant map and $\bar{\nu}$ is the measure on $\overline{B}_{2}^{+}$ given by

[TABLE]

for a positive constant $c_{0}$. Clearly, we have $\Sigma_{\bar{\mu}}=\overline{B}_{2}^{m-4}\times\{0\}$. By definition of $\mathcal{B}_{\Lambda}$ there are maps $u_{i}\in W^{2,2}(B_{2}^{+},N)$ and boundary values $g_{i}\in C^{\infty}(B_{2}^{+},N)$ with $(u_{i},Du_{i})=(g_{i},Dg_{i})$ on $T_{2}$ in the sense of traces, so that

[TABLE]

for any $i\in{\mathds{N}}$, and the following convergence holds as $i\to\infty$.

[TABLE]

In particular, the convergence (46) follows from Lemma 3.2 (i), and the convergence (47) follows from (41).

In the sequel we will use the notation $\mathcal{C}_{r}(x)=\mathcal{C}_{r}(x^{\prime},x^{\prime\prime}):=B_{r}^{m-4}(x^{\prime}){\times}B^{4}_{r}(x^{\prime\prime})$ for cylinders with barycenter $x=(x^{\prime},x^{\prime\prime})\in{\mathds{R}}^{m-4}{\times}{\mathds{R}}^{4}$, and $\mathcal{C}_{r}^{+}(x):=C_{r}(x)\cap{\mathds{R}}^{m}_{+}$. The barycenter will be omitted in the notation if it is zero. For the proof of the theorem we will either construct a smooth biharmonic map ${\mathds{R}}^{4}\to N$ or a biharmonic map ${\mathds{R}}^{4}_{+}\to N$ with constant boundary values as the limit of a blow-up sequence

[TABLE]

where the parameters $p_{i}$ and $\delta_{i}$ will be carefully chosen below. The above map is defined for $y\in\mathcal{C}_{R_{i}}$ with $y_{m}\geq-\delta_{i}^{-1}p_{i,m}$, where $R_{i}:=1/8\delta_{i}\to\infty$, and we write $p_{i,m}$ for the $m$-th component of the vector $p_{i}\in{\mathds{R}}^{m}$. The construction is carried out in several steps.

*Step 1. *We claim that after extracting a subsequence, there holds

[TABLE]

From the monotonicity formula stated in Theorem 2.2 we infer

[TABLE]

for a.e. $0<\rho<r<1$ and every $a\in B_{1}^{m-4}\times\{0\}$, where we used the abbreviations $\Phi_{u_{i}}$ and $\Psi_{u_{i}}$ introduced in (7) and (8) and $X(x):=x-a$. Moreover, since $g_{i}\to c_{\ast}$ in $C^{4}$, Theorem 2.2 yields $K_{i}\to 0$ and $\chi_{i}\to 0$ in the limit $i\to\infty$. Lemma 3.4 implies that after passing to a subsequence, we have the convergence

[TABLE]

for a.e. $0<\rho<r<1$, where the last identity follows from the particular form of $\bar{\nu}$ and the fact $a\in B_{1}^{m-4}\times\{0\}$. Using the last formula to pass to the limit in (51), we deduce

[TABLE]

Moreover, the Morrey bound (44) implies

[TABLE]

which can be made arbitrarily small by choosing $\rho>0$ small enough. Combining the last two formulae, we arrive at

[TABLE]

for a.e. $r\in(0,1]$, where $X(x)=x-a$. We apply this identity once with $a_{0}:=0$ and once with $a_{k}:=\tfrac{1}{4}e_{k}\in B_{1}^{m-4}\times\{0\}$ for $k\in\{1,\ldots,m-4\}$. In this way, we deduce

[TABLE]

in the limit $i\to\infty$. This yields the claim (50).

Step 2 (Choice of $p_{i}^{\prime}$). We claim that the parameters $p_{i}^{\prime}\in B_{1/4}^{m-4}$ can be chosen with the properties

[TABLE]

and

[TABLE]

For the proof of this claim, we consider the functions $f_{i}\in L^{1}({\mathds{R}}^{m-4})$ defined by

[TABLE]

and $f_{i}(x^{\prime})=0$ otherwise. Since $B_{3/8}^{m-4}\times B_{1/4}^{4}\subset B_{1/2}$, the convergence (50) implies $f_{i}\to 0$ in $L^{1}({\mathds{R}}^{m-4})$, as $i\to\infty$. The weak $L^{1}$-estimate for the Hardy-Littlewood maximal function implies

[TABLE]

for every $\varepsilon_{i}>0$, where we used the abbreviation

[TABLE]

With the choices $\varepsilon_{i}:=2c(m)\|f_{i}\|_{L^{1}}/\mathcal{L}^{m-4}(B_{1/4}^{m-4})\to 0$, as $i\to\infty$, the above inequality implies

[TABLE]

Furthermore, the partial regularity result from Corollary 2.6 implies

[TABLE]

Because of (54) and (55), there are points $p_{i}^{\prime}\in B_{1/4}^{m-4}$ satisfying (53) and $\mathcal{M}f_{i}(p_{i}^{\prime})\leq\varepsilon_{i}\to 0$ as $i\to\infty$. The latter property implies (52), so that the claim of Step 2 is verified.

Step 3 (Choice of the scaling factors $\delta_{i}$). The scaling factors have to be chosen carefully to make sure that the bi-energies of the rescaled maps neither tend to zero nor become unbounded, since we want to obtain a non-constant biharmonic map of finite bi-energy in the limit. In order to preserve a certain energy level during the blow-up, we consider the quantity

[TABLE]

for $\delta\in(0,1]$. We may assume that $p_{i}^{\prime}\to:p_{0}^{\prime}\in\overline{B}_{1/4}^{m-4}$ as $i\to\infty$. We will inductively construct a subsequence $i_{j}\in{\mathds{N}}$, $j\in{\mathds{N}}_{0}$, and a decreasing sequence $\delta_{i_{j}}\searrow 0$ with

[TABLE]

Note that we did not claim anything for the case $j=0$, so we can simply choose $i_{0}=1$ and $\delta_{0}=\frac{1}{2}$. Now suppose that $i_{j-1}\in{\mathds{N}}$ and $\delta_{i_{j-1}}>0$ have already been chosen for some $j\in{\mathds{N}}$. For any $\delta_{*}\in(0,\frac{1}{2}\delta_{i_{j-1}})$, we infer from (45) and (48) that

[TABLE]

where the last estimate holds because of $(p_{0}^{\prime},0)\in\Sigma_{\bar{\mu}}$ and $\mathcal{C}_{\delta_{*}}(p_{0}^{\prime},0)\supset B_{\delta_{*}}(p_{0}^{\prime},0)$. Choosing $i_{j}\in{\mathds{N}}$ large enough, depending on $\delta_{*}$, we can thus achieve

[TABLE]

This fixes the index $i_{j}\in{\mathds{N}}$. On the other hand, we know from the choice of $p_{i}^{\prime}$ in Step 2 that the map $u_{i_{j}}$ is smooth on $A_{\delta}:=\overline{B_{\delta}^{m-4}(p_{i_{j}}^{\prime}){\times}B^{4}_{1/4+\delta}}\cap{\mathds{R}}^{m}_{+}$ if $\delta>0$ is chosen small enough. Consequently,

[TABLE]

if $\delta\in(0,\delta_{*}]$ is chosen small enough in dependence on $i_{j}$. Combining the last two estimates and applying the intermediate value theorem, we deduce the existence of a number $\delta_{i_{j}}\in(0,\frac{1}{2}\delta_{i_{j-1}})$ with the property (56). This construction yields the desired sequence $\delta_{i_{j}}\searrow 0$ for $j\to\infty$ with (56), which concludes Step 3. In what follows, we will denote the subsequence $\{i_{j}\}$ again by $\{i\}$ for simplicity.

Step 4 (Choice of $p_{i}^{\prime\prime}$). We choose points $p_{i}^{\prime\prime}\in\overline{B_{1/4}^{4}}\cap{\mathds{R}}^{4}_{+}$ for which $p_{i}:=(p_{i}^{\prime},p_{i}^{\prime\prime})$ satisfies

[TABLE]

for all $i\in{\mathds{N}}$, which is possible by (56) and the definition of $\mathds{F}_{i}$. We claim that for all but finitely many values of $i\in{\mathds{N}}$, we have $p_{i}^{\prime\prime}\in B_{1/8}^{4}$. Indeed, if this was not the case, after passing to a subsequence we would have

[TABLE]

We consider a radius $R_{0}\leq\frac{1}{20}$ to be fixed later independently of $i\in{\mathds{N}}$. We can assume $\sqrt{2}\delta_{i}\leq\frac{R_{0}}{4}$ by choosing $i\in{\mathds{N}}$ large enough. We write $p_{i}^{0}$ for the orthogonal projection of $p_{i}$ onto $\partial{\mathds{R}}^{m}_{+}$ and distinguish between the cases $|p_{i}-p_{i}^{0}|<\frac{R_{0}}{4}$ and $|p_{i}-p_{i}^{0}|\geq\frac{R_{0}}{4}$. In the first case, we observe that $\mathcal{C}^{+}_{\delta_{i}}(p_{i})\subset B^{+}_{\sqrt{2}\delta_{i}}(p_{i})\subset B^{+}_{R_{0}/2}(p_{i}^{0})$. Applying Lemma 2.3 with the center $a=p_{i}^{0}\in T_{1}$, we thus infer

[TABLE]

where we used the property $R_{0}\leq\frac{1}{20}$ in the last step. In the remaining case $|p_{i}-p_{i}^{0}|\geq\frac{R_{0}}{4}$, we again assume $\sqrt{2}\delta_{i}\leq\frac{R_{0}}{4}$, which implies $\mathcal{C}_{\delta_{i}}^{+}(p_{i})\subset B_{\sqrt{2}\delta_{i}}(p_{i})\subset B_{R_{0}/4}(p_{i})$. Therefore, Lemma 2.3 implies

[TABLE]

Now, we recall the choice of $p_{i}^{\prime\prime}$, use either (58) or (59) and then the convergences (45) and (48). In this way, we deduce

[TABLE]

By choosing $R_{0}$ so small that $cR_{0}<2^{1-m}\varepsilon_{0}$, we arrive at the desired contradiction. This yields the claim $p_{i}^{\prime\prime}\in B_{1/8}^{4}$.

Step 5 (Blow-up). As before, we denote the $m$-th component of the vector $p_{i}\in{\mathds{R}}^{m}$ by $p_{i,m}$. We distinguish between the case $\delta_{i}^{-1}p_{i,m}\to\infty$ and the case $\delta_{i}^{-1}p_{i,m}\to b$ for some $b\in[0,\infty)$ in the limit $i\to\infty$. By passing to a subsequence, we can ensure that one of these two alternatives is satisfied. In the first case, we define rescaled maps $v_{i}$ according to

[TABLE]

where $R_{i}:=1/8\delta_{i}\to\infty$. In the case $\delta_{i}^{-1}p_{i,m}\to\infty$, the domain of definition of $v_{i}$ contains arbitrarily large balls centered in the origin. Therefore, this case can be treated analogously as in the interior situation, cf. the arguments following (3.24) in [21]. In this way, it is possible to show that the limit map of the rescaled maps $v_{i}$ is of the form $\hat{v}(x^{\prime},x^{\prime\prime})=v(x^{\prime\prime})$ for a non-constant biharmonic map $v\in C^{\infty}({\mathds{R}}^{4},N)$ with $|D^{2}u|\in L^{2}({\mathds{R}}^{4})$. Under stereographic projection, this map corresponds to a non-constant Paneitz-biharmonic map $v\in C^{\infty}(S^{4},N)$, which is a contradiction to our assumptions.

We omit the details, which have been carried out in [21], and present only the corresponding arguments in the boundary situation, i.e. in the case $\delta_{i}^{-1}p_{i,m}\to b<\infty$, as $i\to\infty$. In this situation, we rescale around the orthogonal projections $p_{i}^{0}\in\partial{\mathds{R}}^{m}_{+}$ of $p_{i}$, i.e. we define maps $v_{i}\in W^{2,2}(\mathcal{C}_{R_{i}}^{+},N)$ by letting

[TABLE]

where $R_{i}:=1/8\delta_{i}\to\infty$. For the corresponding rescaling of the boundary values, we write

[TABLE]

We can assume in particular that $\delta_{i}^{-1}p_{i,m}<R_{i}$ for all $i\in{\mathds{N}}$. As a consequence of our construction, the maps $v_{i}$ have the following properties. By (52) and the fact $(p_{i}^{0})^{\prime\prime}\in B_{1/8}^{4}$, there holds

[TABLE]

as $i\to\infty$, for every $R>0$. Moreover, (57) and the definition of $\mathds{F}_{i}$ implies

[TABLE]

for all $i\in{\mathds{N}}$. Finally, from (44) we infer

[TABLE]

for all $i\in{\mathds{N}}$. From (62) we deduce by Rellich’s theorem, combined with a diagonal sequence argument, that there is a limit map $\hat{v}\in W^{1,2}_{\mathrm{loc}}({\mathds{R}}^{m}_{+},N)$ with $v_{i}\to\hat{v}$ strongly in $W^{1,2}(B_{R}^{+},N)$ for every $R>0$ and almost everywhere, after passing to a subsequence. For a fixed $R>0$, we may additionally find a subsequence $\{i_{j}\}\subset{\mathds{N}}$ with $v_{i_{j}}\rightharpoondown\hat{v}$ weakly in $W^{2,2}(B_{R}^{+},{\mathds{R}}^{L})$. Therefore, (60) implies for every $1\leq k\leq m-4$

[TABLE]

As a consequence, $\partial_{k}\hat{v}$ is constant, and since $N$ is compact, even $\partial_{k}\hat{v}\equiv 0$ for every $1\leq k\leq m-4$. This implies that there is a map $v\in W^{2,2}_{\textrm{loc}}({\mathds{R}}^{4}_{+},N)$ with $\hat{v}(y^{\prime},y^{\prime\prime})=v(y^{\prime\prime})$ for a.e. $(y^{\prime},y^{\prime\prime})\in{\mathds{R}}^{m-4}\times{\mathds{R}}^{4}_{+}$. Furthermore, we recall for further reference that the strong convergence $v_{i}\to\hat{v}$ in $W^{1,2}(B_{R}^{+},N)$ implies

[TABLE]

*Step 6 ($C^{3}$-convergence). * In order to establish local $C^{3}$-convergence $v_{i}\to\hat{v}$, we will show that the bi-energy of the maps $v_{i}$ is small on every cylinder $\mathcal{C}_{1/2}(a)$ for $a\in B_{R}^{+}$, if $i>i_{0}(R)$ is sufficiently large. More precisely, we will show that the quantities

[TABLE]

are small, where $a\in\mathcal{C}^{+}_{R_{i}-1}$ and $\psi\in C^{\infty}_{\mathrm{cpt}}(\mathcal{C}_{1},[0,1])$ is a cut-off function with $\psi\equiv 1$ on $\mathcal{C}_{1/2}$ and $|D^{2}\psi|+|D\psi|\leq c$ for some constant $c=c(m)$. For $1\leq k\leq m-4$, we will use the test vector field $\xi:=\psi_{a}e_{k}$, where $\psi_{a}(y):=\psi(y-a)$, in the differential equation (5) for the maps $v_{i}$. Note that the maps $v_{i}$ also satisfy (5) by scaling invariance, and that the test vector field is admissible since $e_{k}$ is tangential to $\partial{\mathds{R}}^{m}_{+}$. Before applying the differential equation, we calculate

[TABLE]

using integration by parts in the last step. We re-write the first term in the last integral by an application of the differential equation (5) for the maps $v_{i}$, with the result

[TABLE]

The right-hand side vanishes in the limit $i\to\infty$ because the sequence $v_{i}$ is bounded in $W^{2,2}(\mathcal{C}_{1}^{+}(a))$ by (62) and we have the convergences (60), (64), and $h_{i}\to c_{\ast}$ in $C^{4}(\mathcal{C}_{1}^{+}(a))$. Therefore, the above estimate implies $\frac{\partial}{\partial a_{k}}\,\mathcal{F}_{i}\to 0$ uniformly on $B_{R}^{+}$ for every $R>0$ and every $1\leq k\leq m-4$, as $i\to\infty$. Since we know furthermore $\mathcal{F}_{i}((0,a^{\prime\prime}))\leq 2^{1-m}\varepsilon_{0}$ for all $a^{\prime\prime}\in B_{R_{i}}^{4}\cap{\mathds{R}}^{4}_{+}$ by (61), we arrive at

[TABLE]

for all $a\in B_{R}^{+}$, if $i\geq i_{0}(R)$ is chosen sufficiently large. Applying Corollary 2.8 on $B_{1/2}^{+}(a)\subset\mathcal{C}_{1/2}^{+}(a)$, we infer the bound

[TABLE]

By Arzéla-Ascoli, this implies convergence $v_{i}\to\hat{v}$ in $C^{3}(B_{R}^{+},N)$. Since we have already established almost everywhere convergence $v_{i}\to\hat{v}$, it is not necessary to pass to another subsequence. Therefore, the $C^{3}$-convergence holds on every ball $B_{R}^{+}$ with $R>0$.

*Step 7 (Conclusion). * Keeping in mind that $\delta_{i}^{-1}p_{i,m}\to b\in[0,\infty)$ as $i\to\infty$, we deduce from the $C^{3}$-convergence and the identity (61) that

[TABLE]

Consequently, the map $\hat{v}$, and thus also the restriction $v=\hat{v}|_{\{0\}{\times}{\mathds{R}}^{4}_{+}}$, is not constant. On the other hand, (62) implies for any $R>0$

[TABLE]

which yields $D^{2}v\in L^{2}({\mathds{R}}^{4}_{+},N)$. Since the maps $v_{i}$ are weakly biharmonic and converge to $\hat{v}$ in $C^{3}_{\mathrm{loc}}({\mathds{R}}^{m}_{+},N)$, the map $\hat{v}$ is biharmonic as well, and so is its restriction $v=\hat{v}|_{\{0\}{\times}{\mathds{R}}^{4}_{+}}$. As a consequence, we even have $v\in C^{\infty}({\mathds{R}}^{4}_{+},N)$, cf. [8]. Thus, $v\in C^{\infty}({\mathds{R}}^{4}_{+},N)$ is a non-constant biharmonic map with finite bi-energy and constant boundary values on $\partial{\mathds{R}}^{4}_{+}$. By stereographic projection and the conformal invariance of the Paneitz-bi-energy, this corresponds to a smooth, non-constant Paneitz-biharmonic $4$-halfsphere with constant boundary values, cf. Lemma 4.3 (ii). This completes the proof of the theorem. ∎

3.4. Proof of Theorem 1.6

We consider a sequence $u_{i}\in W^{2,2}(B_{4}^{+},N)$, $i\in{\mathds{N}}$, of variationally biharmonic maps with respect to Dirichlet values $g_{i}\in C^{\infty}(B_{4}^{+},N)$ on $T_{4}$ for which

[TABLE]

holds true, for some $\alpha\in(0,1)$. In view of Lemma 2.3, this estimate implies the uniform Morrey space bound

[TABLE]

This means that there exists a constant $\Lambda\geq 1$ so that $(u_{i},0)\in\mathcal{B}_{\Lambda}$ for all $i\in{\mathds{N}}$. Therefore, after passing to a subsequence we have convergence $(u_{i},0)\rightrightarrows(u,\nu)\in\mathcal{B}_{\Lambda}$ in the sense of Definition 3.1, in the limit $i\to\infty$. Let us assume for contradiction that there is no strong subconvergence $u_{i}\to u$ in $W^{2,2}(B_{1}^{+},{\mathds{R}}^{L})$. Then Lemma 3.2 (ii) implies $~{}\operatorname{spt}\nu\cap\overline{B}_{1}^{+}\neq\varnothing$. Therefore, Theorem 3.6 yields the existence of a non-constant Paneitz-biharmonic $4$-sphere or non-constant Paneitz-biharmonic $4$-halfsphere with constant boundary values. But the existence of such maps is excluded by the assumptions of Theorem 1.6, so that the claimed strong convergence holds true. $\square$

4. Liouville type theorems for biharmonic maps on a half

space

The next theorem excludes the existence of certain non-constant biharmonic maps which might occur as tangent maps in singular boundary points. We remark that since these maps are homogeneous of degree zero, they can also be interpreted as maps $v:S^{m-1}_{+}\to N$ that are biharmonic with respect to a certain Paneitz-bi-energy, cf. [21, Lemma 5.1]. We note that the case $m=5$, which corresponds to Paneitz-biharmonic $4$-halfspheres, can not be treated with the same methods and will be postponed to Lemma 4.3.

Theorem 4.1**.**

Let $m\geq 6$ and assume that $u\in W^{2,2}_{\mathrm{loc}}({\mathds{R}}^{m}_{+},N)$ is a stationary biharmonic map that is homogeneous of degree zero and admits constant Dirichlet boundary values in the sense $(u,Du)=(c,0)$ on $\partial{\mathds{R}}^{m}_{+}$. Moreover, we assume that $u$ is smooth in a neighbourhood of $\partial{\mathds{R}}^{m}_{+}\setminus\{0\}$. Then $u$ is constant in ${\mathds{R}}^{m}_{+}$.

Proof.

We consider the vector field $\xi\in C^{\infty}({\mathds{R}}^{m}_{+},{\mathds{R}}^{m})$ defined by $\xi(x):=\eta(|x|)e_{m}$ for a function $\eta\in C^{\infty}_{0}((0,\infty),[0,\infty))$. Since this vector field does not vanish on $\partial{\mathds{R}}^{m}_{+}$, it is not admissible in the differential equation (2). However, because $u$ is smooth in a neighbourhood of $\partial{\mathds{R}}^{m}_{+}\setminus\{0\}$, we infer from Gauß’ theorem that

[TABLE]

where we used the convention to sum all double indices from $1$ to $m$. Using the definition of $\xi$ and the fact $Du=0$ on $\partial{\mathds{R}}^{m}_{+}$, we compute

[TABLE]

In the last step, we used the fact that $u$ is constant on $\partial{\mathds{R}}^{m}_{+}$, which implies $\Delta u=\partial^{2}_{m}u$ on $\partial{\mathds{R}}^{m}_{+}$. Moreover, with the abbreviation $r=|x|$, the definition of $\xi$ implies

[TABLE]

Since $u$ is homogeneous of degree zero, the derivative $\partial_{m}u$ is homogeneous of degree $-1$, which implies $x_{i}\partial_{i}\partial_{m}u=-\partial_{m}u$. By the homogeneity of $u$, we thus infer

[TABLE]

Using integration by parts, we re-write the term $\mathrm{I}_{2}$ to

[TABLE]

At this point, we used the assumption that $\eta$ has compact support in $(0,\infty)$, which implies that the boundary terms vanish. Finally, we compute

[TABLE]

Plugging (4), (67), (68), and (69) into (4), we deduce

[TABLE]

Using the homogeneity of $u$ and Gauß’ theorem, we furthermore compute

[TABLE]

where the boundary integrals vanish because of $\frac{\partial u}{\partial r}=0$ on $S_{1}^{+}$ and $\partial_{m}u=0$ on $\partial{\mathds{R}}^{m}_{+}$. Another application of Gauß’ theorem implies

[TABLE]

We use this identity in (70) and integrate by parts with respect to $r$. In this way, we arrive at

[TABLE]

Since $m>5$ and $\eta\geq 0$, we infer $|Du|=0$ on $S_{1}^{+}$. By the homogeneity of $u$, this implies that $u$ is constant on ${\mathds{R}}^{m}_{+}$, as claimed. ∎

Remark 4.2**.**

For further reference we note that in the case $m=5$, the preceding proof yields that $D^{2}u\equiv 0$ holds true on $\partial{\mathds{R}}^{5}_{+}$ for any map $u\in W^{2,2}_{\mathrm{loc}}({\mathds{R}}^{5}_{+},N)$ as in Theorem 4.1. For the verification of this assertion, we observe that the condition $m>5$ was only used in the very last step of the proof, so that identity (71) also holds in the case $m=5$. This identity implies $\partial_{5}^{2}u\equiv 0$ on $\partial{\mathds{R}}^{5}_{+}$, and since $Du\equiv 0$ holds on $\partial{\mathds{R}}^{5}_{+}$ by assumption, we deduce that all second derivatives vanish on the boundary. **

In the case $m=5$, we obtain the result corresponding to Theorem 4.1 under the additional assumption of non-existence of Paneitz-biharmonic $4$-halfspheres.

Lemma 4.3**.**

Assume that the manifold $N$ does not carry any non-constant Paneitz-biharmonic $4$-halfspheres with constant boundary values. Then the following two statements holds true.

1. (i)

Any biharmonic map $u\in C^{\infty}({\mathds{R}}^{5}_{+}\setminus\{0\},N)$ that is homogeneous of degree zero and attains constant boundary values $(u,Du)=(c,0)$ on $\partial{\mathds{R}}^{5}_{+}$ is constant. 2. (ii)

Any biharmonic map $w\in C^{\infty}({\mathds{R}}^{4}_{+},N)$ with finite bi-energy and constant boundary values $(w,Dw)=(c,0)$ on $\partial{\mathds{R}}^{4}_{+}$ is constant.

Proof.

(i) Since $u$ is homogeneous of degree zero, we have

[TABLE]

in ${\mathds{R}}^{5}_{+}\setminus\{0\}$, where $\Delta_{S}$ denotes the Laplace-Beltrami operator on $S^{4}_{+}$, and we abbreviated $r=|x|$. Since $u$ is biharmonic on ${\mathds{R}}^{5}_{+}\setminus\{0\}$, we deduce that the restriction $v:=u|_{S^{4}_{+}}$ is Paneitz-biharmonic on $S^{4}_{+}$. Hence, the map $v$ is constant by assumption, and so is its homogeneous extension $u$.

(ii) By stereographic projection, the biharmonic map $w\in C^{\infty}({\mathds{R}}^{4},N)$ gives rise to a map $v\in C^{\infty}(S^{4}_{+}\setminus\{e_{1}\},N)$. The conformal invariance of the Paneitz-bi-energy, see [16, 4], implies that

[TABLE]

and that $v\in W^{2,2}(S^{4}_{+},N)$ is a critical point of the Paneitz-bi-energy. More precisely, the map $v$ is Paneitz-biharmonic on $S^{4}_{+}\setminus\{e_{1}\}$. To conclude, it remains to show that the singularity in $e_{1}$ can be removed. To this end, we consider the homogeneous extension $u(x):=v(\frac{x}{|x|})$. The identity (72) implies that $u\in W^{2,2}_{\mathrm{loc}}({\mathds{R}}^{5}_{+},N)$ is biharmonic on ${\mathds{R}}^{5}_{+}\setminus({\mathds{R}}_{\geq 0}e_{1})$. Since ${\mathds{R}}_{\geq 0}e_{1}$ is a one-dimensional set and is therefore negligible with respect to the $W^{2,2}$-capacity, we infer that $u$ is weakly biharmonic on ${\mathds{R}}^{5}_{+}$. Moreover, since $u$ is the homogeneous extension of a map in $W^{2,2}\cap W^{1,4}(S^{4}_{+},N)$, for any $\varepsilon>0$ we can choose a radius $\delta>0$ sufficiently small to ensure

[TABLE]

Therefore, the $\varepsilon$-regularity result from [7, Lemma 3.1] can be applied, which yields that $v$ is smooth in a neighbourhood of $e_{1}$. We point out that in view of (73), no monotonicity formula is required for this result. Consequently, we have shown that $v\in C^{\infty}(S^{4}_{+},N)$ is a Paneitz-biharmonic map with constant Dirichlet boundary values. By assumption, the map $v$ is constant, and so is the map $w\in C^{\infty}({\mathds{R}}^{4}_{+},N)$. This completes the proof of the lemma. ∎

We end this section with an example of a target manifold for which our standing assumptions on the non-existence of Paneitz-biharmonic spheres are satisfied.

Proposition 4.4**.**

Let $N=S_{r_{1}}^{1}\times\ldots\times S_{r_{n}}^{1}\subset{\mathds{R}}^{2n}$ be a flat torus, where $n\geq 1$ and $r_{i}>0$ for $i=1,\ldots,n$. Then $N$ does neither carry a non-constant Paneitz-biharmonic $4$-sphere nor a non-constant Paneitz-biharmonic $4$-halfsphere with constant boundary values.

Proof.

The assertion concerning the full spheres has already been proved in [21, Lemma 5.4]. Here, we demonstrate that the same argument can be applied to yield the constancy of the corresponding halfspheres with constant boundary values.

To this end, we consider a Paneitz-biharmonic map $u\in C^{\infty}(S^{4}_{+},N)$ with constant boundary values in the sense $(u,Du)=(c,0)$ on $\partial S^{4}_{+}$. We extend $u$ to a function on ${\mathds{R}}^{5}_{+}\setminus\{0\}$ that is homogeneous of degree zero. Because of identity (72), the extended function, which we still denote by $u$, is biharmonic on ${\mathds{R}}^{5}_{+}\setminus\{0\}$. Therefore, Remark 4.2 implies $D^{2}u\equiv 0$ on $\partial{\mathds{R}}^{5}_{+}$. For every vector field $V\in C^{\infty}(S^{4}_{+},{\mathds{R}}^{2n})$ with $V(x)\in T_{u(x)}N$ for every $x\in S^{4}_{+}$, the fact $\Delta_{S}^{2}u(x)-2\Delta_{S}u(x)\perp T_{u(x)}N$ for all $x\in S^{4}_{+}$ and two integrations by parts imply

[TABLE]

Since $\Delta_{S}u=0$ on $\partial S^{4}_{+}$, the boundary integral vanishes for every test vector field with $V\equiv 0$ on $\partial S^{4}_{+}$. In particular, this holds true for the choice $V:=\Delta_{S}^{\top}u$, where $\Delta_{S}^{\top}$ denotes the tangential part of the Laplace-Beltrami operator on $S^{4}_{+}$. We assume that $V$ is extended to a function on ${\mathds{R}}^{5}_{+}\setminus\{0\}$ that is homogeneous of degree zero. More precisely, this means that $V(x)=|x|^{2}\nabla^{u}_{e_{i}}\partial_{i}u(x)$ for any $x\in{\mathds{R}}^{5}_{+}\setminus\{0\}$, where we used the notation $\nabla^{u}$ for the covariant derivative on the bundle $u^{\ast}\mathrm{T}N$. Since $u$ and $V$ are both homogeneous of degree zero, we have

[TABLE]

and

[TABLE]

where $A$ denotes the second fundamental form of the submanifold $N\subset{\mathds{R}}^{2n}$. Moreover, we employ the usual summation convention and sum all double indices from $1$ to $5$. Consequently, choosing $V:=\Delta_{S}^{\top}u$ in (74), we infer

[TABLE]

In the first and the third term appearing on the right-hand side, we use the identity

[TABLE]

which follows from the definition of the covariant derivative and the second fundamental form. Next, using the equations by Weingarten and Gauß and keeping in mind that the Riemannian curvature of $N$ vanishes, we deduce

[TABLE]

Finally, because the flat torus has a parallel second fundamental form, i.e. $\nabla^{\perp}A\equiv 0$, we can compute

[TABLE]

Using the preceding observations in (75), we deduce

[TABLE]

Next, we use the identity $\nabla^{u}_{e_{k}}\nabla^{u}_{e_{k}}V=\operatorname{tr}_{S}(\nabla^{u}\nabla^{u}V)$ on $S^{4}_{+}$, where we abbreviated $\operatorname{tr}_{S}$ for the trace on $\mathrm{T}S^{4}_{+}$. Integrating by parts and recalling the definition of $V$, we then compute

[TABLE]

Since $N$ has vanishing Riemannian curvature, we have the following identity on $S^{4}_{+}$.

[TABLE]

where the last step follows from the fact that $\partial_{k}u$ is homogeneous of degree $-1$. We use this identity, recall the fact $x_{k}\partial_{k}u=0$ and integrate by parts. Since the second fundamental form is parallel, we infer

[TABLE]

We note that the boundary term appearing in the integration by parts vanishes because of $Du\equiv 0$ on $\partial{\mathds{R}}^{5}_{+}$. The second integral on the right-hand side is non-positive, and so is the first one, because the Gauß equations on the flat manifold $N$ imply

[TABLE]

Consequently, equation (78) implies

[TABLE]

Using (77) and (79) in (76), we arrive at

[TABLE]

This implies that both the tangential and the normal part of $\Delta u$ vanish on $S^{4}_{+}$, which means that $u:{\mathds{R}}^{5}_{+}\setminus\{0\}\to{\mathds{R}}^{2n}$ is a harmonic function. Classical regularity results for harmonic functions now yield $u\in C^{\infty}({\mathds{R}}^{5}_{+},N)$. Since $u$ is homogeneous of degree zero, this implies that it is a constant map, as claimed. This completes the proof of the proposition. ∎

5. Full boundary regularity

5.1. Properties of tangent maps

The strategy for the proof of the full boundary regularity is to apply the dimension reduction argument by Federer in order to prove that the dimension of the singular set is zero. To this end, we first provide some properties of tangent maps of a variationally biharmonic map $u\in W^{2,2}(B^{+}_{2},N)$ in a boundary point $a\in T_{1}$. For $r\in(0,1)$ we define the rescaled maps

[TABLE]

A map $v\in W^{2,2}_{\mathrm{loc}}({\mathds{R}}^{m}_{+},N)$ is called a tangent map of $u$ at the point $a\in T_{1}$ if there is a sequence $r_{i}\searrow 0$ with $u_{a,r_{i}}\to v$ strongly in $W^{2,2}_{\mathrm{loc}}({\mathds{R}}^{m},N)$ as $i\to\infty$.

The Compactness Theorem 1.6 is crucial at this point to ensure the strong convergence of suitable rescaled maps to a tangent map. Moreover, we obtain the following structure theorem for tangent maps of biharmonic maps.

Theorem 5.1**.**

Assume that $u\in W^{2,2}(B_{2}^{+},N)$ is a weakly biharmonic map that satisfies the stationarity condition (5) and satisfies $(u,Du)=(g,Dg)$ on $T_{2}$ in the sense of traces. Moreover, we assume that $N$ does not carry any nontrivial Paneitz-biharmonic $4$-spheres nor Paneitz-biharmonic $4$-halfspheres with constant boundary data. Then the following statements are true:

1. (i)

For any $a\in T_{1}$ and any sequence $r_{i}\searrow 0$ there is a tangent map $v\in W^{2,2}_{loc}({\mathds{R}}^{m}_{+},N)$ so that $u_{a,r_{i}}$ subconverges to $v$ in $W^{2,2}_{loc}({\mathds{R}}^{m}_{+},N)$ as $i\to\infty$. The tangent map is again weakly biharmonic, satisfies (5) and attains constant Dirichlet boundary values on $\partial{\mathds{R}}^{m}_{+}$. 2. (ii)

Every tangent map of $u$ in a point $a\in T_{1}$ is homogeneous of degree zero. 3. (iii)

Let $s\geq 0$. At $\mathcal{H}^{s}$-a.e. point $a\in\operatorname{sing}(u)\cap T_{1}$ there exists a tangent map $v$ of $u$ with $\mathcal{H}^{s}(\operatorname{sing}(v)\cap T_{1})>0$.

Proof.

The strong subconvergence $u_{a,r_{i}}\to v$ to some tangent map $v\in W^{2,2}_{\mathrm{loc}}({\mathds{R}}^{m}_{+},N)$ is a consequence of the Compactness Theorem 1.6. The differential equations (1) and (5) are preserved under strong convergence, which yields the statements on the biharmonicity of the tangent map. The boundary values of the rescaled maps $u_{a,r_{i}}$ are given by the maps $g_{a,r_{i}}$, which satisfy

[TABLE]

This implies $g_{a,r_{i}}\to c$ for a constant $c\in N$ and proves that the tangent map attains constant boundary values. This completes the proof of (i).

Our next goal is to show (ii). To this end, let $v$ be a tangent map of $u$ in a point $a\in T_{1}$. Since $v$ possesses constant boundary values, the monotonicity formula from Theorem 2.2 with center $a=0$ reads

[TABLE]

for a.e. $0<\rho<r$, where $X(x)=x$. The strong convergence $u_{a,r_{i}}\to v$ implies that the functions $f_{i}(\rho):=\Phi_{u_{a,r_{i}}}(0,\rho)$ converge in $L^{1}_{\mathrm{loc}}([0,\infty))$ to $f(\rho):=\Phi_{v}(0,\rho)$, as $i\to\infty$. Therefore, for any $R>0$ we deduce

[TABLE]

since the last limit exists by Corollary 2.4. We deduce that the left-hand side does not depend on $R>0$. Therefore, inequality (80) implies

[TABLE]

for any $R>0$, which is only possible if $\partial_{X}v=0$. This yields the asserted homogeneity of the tangent map.

For the proof of the last statement (iii), we recall a standard result on densities of sets, which states that for $\mathcal{H}^{s}$-a.e. point $a\in\operatorname{sing}(u)\cap T_{1}$ there is a sequence $r_{i}\searrow 0$ with

[TABLE]

where $\alpha(s):=\Gamma(\frac{1}{2})^{s}/\Gamma(\frac{s}{2}+1)$, cf. Remark 3.7 in [26]. As shown above, after passing to a subsequence, the rescaled maps $u_{a,r_{i}}$ converge to a tangent map $v\in W^{2,2}_{\mathrm{loc}}({\mathds{R}}^{m}_{+},N)$ locally with respect to the $W^{2,2}$-norm. Let us assume for contradiction that $\mathcal{H}^{s}(\operatorname{sing}(v)\cap T_{1})=0$. Then for any $\varepsilon>0$ there is a cover $\cup_{j\in{\mathds{N}}}B_{j}\supset\operatorname{sing}(v)\cap T_{1}$ of open balls $B_{j}$ with radii $\rho_{j}$ so that

[TABLE]

We claim that $\operatorname{sing}(u_{a,r_{i}})\cap T_{1}\subset U_{\varepsilon}:=\cup_{j}B_{j}$ for all but finitely many values of $i\in{\mathds{N}}$. If this was not the case, after passing to a subsequence we could choose singular boundary points $p_{i}\in\operatorname{sing}(u_{a,r_{i}})\cap T_{1}$ with $p_{i}\to p\not\in\operatorname{sing}(v)$ as $i\to\infty$. Since $v$ is smooth in a neighbourhood of $p$, we have

[TABLE]

for all sufficiently small radii $r\in(0,\delta)$, where $\varepsilon_{1}>0$ denotes the constant from the Regularity Theorem 2.5 for the case of constant boundary values. The preceding inequality implies for sufficiently large values of $i>i_{0}(r)$ that

[TABLE]

holds true, since we can in particular assume $B_{r/2}^{+}(p_{i})\subset B_{r}^{+}(p)$. Theorem 2.5 therefore implies $p_{i}\not\in\operatorname{sing}(u_{a,r_{i}})$, which is a contradiction. We conclude that $\operatorname{sing}(u_{a,r_{i}})\cap T_{1}\subset U_{\varepsilon}$ holds true for all sufficiently large $i\in{\mathds{N}}$. But this means

[TABLE]

for all sufficiently large $i\in{\mathds{N}}$, where we used (82) in the second last step. By choosing $\varepsilon:=2^{-s-1}$, we achieve a contradiction to the density condition (81), which completes the proof of the theorem. ∎

5.2. Proof of Theorem 1.8

Having established the properties of the tangent maps gathered in the preceding theorem, the proof of Theorem 1.8 follows in a standard way from the dimension reduction principle due to Federer, see for example [26], Thm. A.4. We briefly sketch the remainder of the proof.

Let $d\in{\mathds{N}}_{0}$ be the smallest natural number with the property that

[TABLE]

holds true for every weakly biharmonic map $u\in W^{2,2}(B_{1}^{+},N)$ with (5) and smooth boundary values $g\in C^{\infty}(B_{1}^{+},N)$ on $T_{1}$. Here, $\mbox{dim}_{\mathcal{H}}$ denotes the Hausdorff dimension. We assume for contradiction that there are weakly biharmonic maps with (5) and smooth boundary values that have singular boundary points. Then by definition of $d$, we can find such a weakly biharmonic map $u\in W^{2,2}(B_{1}^{+},N)$ and some $s\in(d-1,d]$ with $\mathcal{H}^{s}(\operatorname{sing}u\cap T_{1})>0$. By Theorem 5.1 there exists a tangent map $v_{0}$ of $u$ in a singular boundary point $a\in\operatorname{sing}u\cap T_{1}$ that satisfies $\mathcal{H}^{s}(\operatorname{sing}v_{0}\cap T_{1})>0$ as well. Moreover, $v_{0}$ possesses constant boundary values and is homogeneous of degree zero. Unfortunately, we do not know whether $v_{0}$ is smooth in a neighbourhood of $\partial{\mathds{R}}^{m}_{+}\setminus\{0\}$, so that Theorem 4.1 is not applicable. Therefore, in the case $d>0$, we repeat the procedure and construct in turn a tangent map $v_{1}$ of $v_{0}$ in a point $b\in\operatorname{sing}v_{0}\cap T_{1}\setminus\{0\}$ that again satisfies $\mathcal{H}^{s}(\operatorname{sing}v_{1}\cap T_{1})>0$. Moreover, this second tangent map is homogeneous of degree zero and satisfies moreover $\partial_{b}v_{1}\equiv 0$. This implies in particular that the singular set contains the one-dimensional subspace ${\mathds{R}}b\subset\partial{\mathds{R}}^{m}_{+}$. We repeat the last step $d$ times to successively construct tangent maps $v_{1},v_{2},\ldots,v_{d}$ whose singular sets contain subspaces of increasingly higher dimension. The result is a tangent map $v_{d}\in W^{2,2}({\mathds{R}}^{m}_{+},N)$ whose singular set contains a $d$-dimensional linear subspace $V\subset\partial{\mathds{R}}^{m}_{+}$. After a rotation, we may assume that $\operatorname{sing}v_{d}\supset{\mathds{R}}^{d}\times\{0\}$. Moreover, by construction, the map $v_{d}$ is weakly biharmonic, satisfies (5), has constant boundary values on $T_{1}$, is homogeneous of degree zero and constant in direction of ${\mathds{R}}^{d}\times\{0\}$. By the definition of $d$ according to (83), we know

[TABLE]

In view of the partial regularity result from Corollary 2.6, this is only possible if $d\leq m-5$. We consider the restriction $\tilde{v}:=v_{d}|_{\{0\}\times{\mathds{R}}^{m-d}_{+}}\in W^{2,2}_{\mathrm{loc}}({\mathds{R}}^{m-d}_{+},N)$. A standard cut-off argument implies that $\tilde{v}$ is stationary biharmonic on ${\mathds{R}}^{m-d}_{+}$, cf. [21, Lemma 4.3]. Moreover, $\tilde{v}$ has constant boundary values in the sense of $(u,Du)=(c,0)$ on $\partial{\mathds{R}}^{m-d}_{+}$, and $\tilde{v}$ is homogeneous of degree zero. Moreover, by construction we have $\operatorname{sing}\tilde{v}\cap\partial{\mathds{R}}^{m-d}_{+}=\{0\}$, which means that $\tilde{v}$ is smooth on a neighbourhood of $\partial{\mathds{R}}^{m-d}_{+}\setminus\{0\}$. In the case $d\leq m-6$, we may therefore apply Theorem 4.1, which implies that $\tilde{v}$ is constant. If $d=m-5$, the partial regularity result from Corollary 2.6 and the homogeneity of $\tilde{v}$ imply that $\tilde{v}$ is smooth on ${\mathds{R}}^{m-d}_{+}$. Therefore, our assumptions and Lemma 4.3 again yield that $\tilde{v}$ is a constant map. In any case, we obtain a contradiction to $\operatorname{sing}(\tilde{v})\cap\partial{\mathds{R}}^{m-d}_{+}=\{0\}$. We conclude that our assumption must have been false, i.e. $\operatorname{sing}u\cap T_{1}=\varnothing$ holds for any weakly biharmonic map $u\in W^{2,2}(B_{1}^{+},N)$ with (5) and smooth boundary values. This completes the proof of Theorem 1.8. ∎