Boundary partial regularity for a class of biharmonic maps

Abstract

We consider the Dirichlet problem for stationary biharmonic maps u from a bounded, smooth domain Ω ⊂ R n (n ≥ 5) to a compact, smooth Riemannian manifold N ⊂ R l without boundary. For any smooth boundary data, we show that if, in addition, u satisfies a certain boundary monotonicity inequality, then there exists a closed subset Σ ⊂ Ω, with H n−4 (Σ) = 0, such that

Boundary regularity for minimizing biharmonic maps

“…Unlike the interior monotonicity formula, the boundary monotonicity formula is an artificial assumptionit is unknown whether it can be deduced for all stationary maps. The result [9] is a biharmonic counterpart of a result by C. Wang [28] for stationary harmonic maps.…”

“…The following theorem is a boundary analogue of the monotonicity formula. It was first proved for any W 4,2 biharmonic map (not necessary minimizing), see [9,Section 2]. Recently a boundary monotonicity formula was derived for all minimizing biharmonic mappings in W 2,2 with sufficiently smooth boundary data, see [2].…”

“…The other result concerns partial boundary regularity for stationary maps. It was shown in [9] by Gong et al that if we impose an additional condition on the boundary mapping then there exists a closed subset Σ ⊆ Ω, with H m−4 (Σ) = 0 such that the stationary biharmonic map is smooth up to the boundary, except possibly the set Σ. The additional condition is the boundary monotonicity formula.…”

Regularity and uniqueness of a class of biharmonic map heat flows

“…It asserts smoothness of biharmonic maps when the dimension n = 4, and the partial regularity of stationary biharmonic maps when n ≥ 5. Here we mention in passing the interesting works on biharmonic maps by Angelsberg [1], Strzelecki [31], Hong-Wang [17], Lamm-Rivière [24], Struwe [40], Ku [20], Gastel-Scheven [10], Scheven [34,35], Lamm-Wang [25], Moser [28,29], Gastel-Zorn [11], Hong-Yin [18], and Gong-Lamm-Wang [12]. Now we describe the initial and boundary value problem for the heat flow of biharmonic maps.…”

Higher order intrinsic weak differentiability and Sobolev spaces between manifolds

“…For every k ∈ N * and every p ∈ [1, +∞), Sobolev maps between the manifolds M and N can be defined by first embedding the target manifold N in a Euclidean space R ν for some ν ∈ N through an isometric embedding ι ∈ C k (N, R ν ) and then considering the set [2,22,38] W k,p ι (M, N ) = u : M → N : ι • u ∈ W k,1 loc (M, R ν ) and |D k (ι • u)| ∈ L p (M ) . (1) Since every Riemannian manifold N can be smoothly isometrically embedded in a Euclidean space [40, theorem 3], this definition (1) is always possible.…”