A Regularity Theory of Biharmonic Maps

In this article we prove the regularity of weakly biharmonic maps of domains in Euclidean four space into spheres, as well as the corresponding partial regularity result of stationary biharmonic maps of higher-dimensional domains into spheres. c © 1999 John Wiley & Sons, Inc. Introduction In this article we consider the notion of biharmonic maps and begin an analytic study of the regularity properties of such maps in dimensions greater than or equal to four. To motivate our study, we observe that the conformal transformations of Euclidean spaces are not in general harmonic except in dimension two. The basic reason is that the energy integrand for harmonic maps is conformally invariant only in dimension two. Thus it is natural to study critical points of the conformally invariant energy functionals. There have been several studies of the energy integrand associated with the p-Laplacian (see, for example, [5, 11]). In dimension n the natural first-order functional is the conformally invariant n-energy: R |∇u|n. Unfortunately, the class of n-harmonic maps, although quite abundant, do not enjoy good regularity properties due to the possible degeneration of |∇u|n (see [9]). For this reason, it is of interest to study higher-order energy functionals that are conformally invariant and enjoy better regularity properties. In this article we consider for simplicity the class of biharmonic maps from Euclidean domains to spheres. We realize the standard spheres Sk as unit vectors in R k+1, and consider maps u : Ω → Sk as vector-valued functions that are contained in Sk. The energy functional for biharmonic maps is then R Ω |∆u|2 dx. A locally defined biharmonic map is a map that is critical with respect to compactly supported variations. We note that in the case where the domain has dimension four, this energy functional is conformally invariant, and hence conformal maps of Euclidean four-space are biharmonic in this sense. We remark that this definition of biharmonic map depends on the embedding of the target space in Euclidean space. We Communications on Pure and Applied Mathematics, Vol. LII, 0001–0025 (1999) c © 1999 John Wiley & Sons, Inc. CCC 0010–3640/99/10001-25 2 S.-Y. A. CHANG, L. WANG, AND P. C. YANG do not use the more natural definition in which the energy integrand is replaced by the intrinsic |(∆u)T |2 where vT denotes the tangential component of the vector vT . In analogy with the regularity theory of harmonic maps, we derive corresponding regularity results for biharmonic maps. Our main results are the following: • Theorem 2.1: Any biharmonic map in W 2,2 defined on a disk of dimension four to the standard sphere Sk is Hölder-continuous. • Theorem 4.1: A stationary biharmonic map from an m-dimensional Euclidean disk (m ≥ 5) to the sphere Sk is Hölder-continuous except on a set of (m−4)–dimensional Hausdorff measure zero. • Theorem 5.1: If u is a weak solution of the biharmonic map equation and if u is continuous in B1, then u is smooth. A companion article [2] to this one provides a simplified treatment of the analogues of the preceding results for harmonic maps and serves as an introduction to the techniques used here as well as references to previous work. Our method builds on the technique first introduced by Hélein [7] to write the nonlinearity in determinant form but proceeds more directly to exploit the special quadratic structure of the nonlinearity; thus we were able to avoid the deep structure theory of Hardy BMO duality. Our argument may allow flexibility to deal with other problems of this kind. We hope to return to the problem involving general targets in a future article. We mention here the related article [1] that proves regularity of minimizing solutions of semilinear scalar equations of fourth order with nonlinearity of similar structure to the biharmonic map equation. We also mention that Hardt and Mou also have some regularity results for locally minimizing biharmonic maps [6]. We remark here that Theorems 2.1 and 4.1 remain valid for maps from domains in a Riemannian manifold. In fact, the elliptic estimates we use remain valid provided we interpret all derivatives in the formula as covariant derivatives. Recently a result analogous to Theorem 2.1 with the extrinsic quantity ∆u replaced by the intrinsic (∆u)T was also established by Y. Ku. 1 Derivation of the Euler Equation Consider u a map (Mm,g) → (Sk,h) with h the standard canonical metric on the unit sphere Sk. Suppose u = (u1, . . . ,uk+1) is a critical point of the energy functional; define E2(u) ≡ R M ∑ k+1 α=1(∆gu α)2 dVg. In this section we will derive the Euler-Lagrange equation for u. PROPOSITION 1.1 Suppose u ∈ W 2,2 is a critical point of the functional E2; then u satisfies ∆2uα = −uαλ , α = 1,2, . . . ,k + 1, (1.1) where λ = ∑ β=1[(∆u β)2 + ∆(|∇uβ|2) + 2∇uβ ·∇∆uβ] and ∇∆uβ exists in the Lp sense for all p < 4 . REGULARITY OF BIHARMONIC MAPS 3 PROOF: Since u : Mm → Sk, the Euler equation of E2(u) = 0 satisfies (∆u) = 0 , where (∆2u)T denotes the tangential component of ∆2u. Therefore for some λ, ∆2uα = (∆2uα)N = −uαλ where (∆2u)N denotes the normal component of ∆2u. It remains to compute λ. To do so, we observe that when the target manifold of the map is Sk, we have uβ · uβ = 1; hence ∇uβ · uβ = 0 and ∆uβ · uβ = −|∇uβ|2 (where we treat uβ as a vector, and the equality holds by summing over β). Thus if we inner product both sides of (1.1) by uα and sum over α, we get k+1 ∑ α=1 ∆2uα ·uα = −λ . (1.2) Multiplying both sides of (1.2) by a testing function φ ∈ C∞ 0 (M) and integrating over M, we get − Z λφ = ∑ α Z (∆2uα)uαφ