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Energy convexity of intrinsic bi-harmonic maps and applications I: Spherical target

Paul Laurain and Longzhi Lin

Abstract

In this paper, we show an energy convexity and thus uniqueness for weakly intrinsic bi-harmonic maps from the unit 4-ball B 1 ⊂ ℝ 4 into the sphere 𝕊 n . In particular, this yields a version of uniqueness of weakly harmonic maps on the unit 4-ball which is new. We also show a version of energy convexity along the intrinsic bi-harmonic map heat flow into 𝕊 n , which in particular yields the long-time existence of the intrinsic bi-harmonic map heat flow, a result that was until now only known assuming the non-positivity of the target manifolds by Lamm [26]. Further, we establish the previously unknown result that the energy convexity along the flow yields uniform convergence of the flow.

A ε-regularity for approximate bi-harmonic maps

First, we recall the main result of Lamm and Rivière [28] that provides a divergence form to elliptic fourth order system of certain type (see e.g. (A.1)) under small energy assumption. This will be one of the main tools in order to obtain the estimates needed for the energy convexity for intrinsic and extrinsic bi-harmonic maps into 𝕊 n , see Theorem A.4. The first three results in this appendix work for any general closed target manifold 𝒩 , viewed as a submanifold of ℝ n + 1 (of any co-dimension).

Proposition A.1.

Equation (1.1) and (1.2) can be rewritten in the form

(A.1) Δ 2 ⁢ u = Δ ⁢ ( V ⁢ ∇ ⁡ u ) + div ⁡ ( w ⁢ ∇ ⁡ u ) + ∇ ⁡ ω ⁢ ∇ ⁡ u + F ⁢ ∇ ⁡ u ,

where we have V ∈ W 1 , 2 ⁢ ( B 1 , M n + 1 ⊗ Λ 1 ⁢ R 4 ) , w ∈ L 2 ⁢ ( B 1 , M n + 1 ) , ω ∈ L 2 ⁢ ( B 1 , so n + 1 ) and F ∈ L 2 ⋅ W 1 , 2 ⁢ ( B 1 , M n + 1 ⊗ Λ 1 ⁢ R 4 ) with

(A.2) | V | ≤ C ⁢ | ∇ ⁡ u | ,
| F | ≤ C ⁢ | ∇ ⁡ u | ⁢ ( | ∇ 2 ⁡ u | + | ∇ ⁡ u | 2 ) ,
| w | + | ω | ≤ C ⁢ ( | ∇ 2 ⁡ u | + | ∇ ⁡ u | 2 ) ,

almost everywhere, where C > 0 is a constant which depends only on N .

Theorem A.2 ([28, Theorem 1.4]).

There exist constants ε > 0 , C > 0 depending only on N such that the following holds: let V ∈ W 1 , 2 ⁢ ( B 1 , M n + 1 ⊗ Λ 1 ⁢ R 4 ) , w ∈ L 2 ⁢ ( B 1 , M n + 1 ) , ω ∈ L 2 ⁢ ( B 1 , so n + 1 ) and F ∈ L 2 ⋅ W 1 , 2 ⁢ ( B 1 , M n + 1 ⊗ Λ 1 ⁢ R 4 ) be such that

∥ V ∥ W 1 , 2 + ∥ w ∥ 2 + ∥ ω ∥ 2 + ∥ F ∥ L 2 ⋅ W 1 , 2 < ε ;

then there exist A ∈ L ∞ ∩ W 2 , 2 ⁢ ( B 1 , G ⁢ l n + 1 ) and B ∈ W 1 , 4 3 ⁢ ( B 1 , M n + 1 ⊗ Λ 2 ⁢ R 4 ) such that

∇ ⁡ Δ ⁢ A + Δ ⁢ A ⁢ V - ∇ ⁡ A ⁢ w + A ⁢ ( ∇ ⁡ ω + F ) = curl ⁡ B ,
∥ A ∥ W 2 , 2 + dist ⁡ ( A , 𝒮 ⁢ 𝒪 n + 1 ) + ∥ B ∥ W 1 , 4 3 ≤ C ⁢ ( ∥ V ∥ W 1 , 2 + ∥ w ∥ 2 + ∥ ω ∥ 2 + ∥ F ∥ L 2 ⋅ W 1 , 2 ) .

Thanks to the previous theorem, we are in position to rewrite equations in approximate form of (A.1) in divergence form.

Theorem A.3 ([28, Theorems 1.3 and 1.5]).

There exist constants ε > 0 and C > 0 depending only on N such that if u ∈ W 2 , 2 ⁢ ( B 2 , R n + 1 ) satisfies

Δ 2 ⁢ u = Δ ⁢ ( V ⁢ ∇ ⁡ u ) + div ⁡ ( w ⁢ ∇ ⁡ u ) + ∇ ⁡ ω ⁢ ∇ ⁡ u + F ⁢ ∇ ⁡ u + f ,

where we have V ∈ W 1 , 2 ⁢ ( B 2 , M n + 1 ⊗ Λ 1 ⁢ R 4 ) , w ∈ L 2 ⁢ ( B 2 , M n + 1 ) , ω ∈ L 2 ⁢ ( B 2 , so n + 1 ) , F ∈ L 2 ⋅ W 1 , 2 ⁢ ( B 2 , M n + 1 ⊗ Λ 1 ⁢ R 4 ) and f ∈ L 1 ⁢ ( B 2 , R n + 1 ) with

∥ V ∥ W 1 , 2 + ∥ w ∥ 2 + ∥ ω ∥ 2 + ∥ F ∥ L 2 ⋅ W 1 , 2 < ε ,

then there exist A ∈ L ∞ ∩ W 2 , 2 ⁢ ( B 1 ⁢ G ⁢ l n + 1 ) and B ∈ W 1 , 4 3 ⁢ ( B 1 , M n + 1 ⊗ Λ 2 ⁢ R 4 ) such that

(A.3) ∥ A ∥ W 2 , 2 + d ⁢ ( A , 𝒮 ⁢ 𝒪 n + 1 ) + ∥ B ∥ W 1 , 4 3
  ≤ C ⁢ ( ∥ V ∥ W 1 , 2 + ∥ w ∥ 2 + ∥ ω ∥ 2 + ∥ F ∥ L 2 ⋅ W 1 , 2 )

and

Δ ⁢ ( A ⁢ Δ ⁢ u ) = div ⁡ ( 2 ⁢ ∇ ⁡ A ⁢ Δ ⁢ u - Δ ⁢ A ⁢ ∇ ⁡ u + A ⁢ w ⁢ ∇ ⁡ u + ∇ ⁡ A ⁢ ( V ⁢ ∇ ⁡ u ) - A ⁢ ∇ ⁡ ( V ⁢ ∇ ⁡ u ) - B ⁢ ∇ ⁡ u ) + A ⁢ f .

A first consequence of Theorem A.3, is the ε-regularity for approximate intrinsic and extrinsic bi-harmonic maps into 𝕊 n . This is a refined version of [31, Theorem 2.3], since here we only assume the smallness on the bi-energy (rather than ∥ ∇ 2 ⁡ u ∥ 2 2 + ∥ ∇ ⁡ u ∥ 4 4 ) and use the fact that the map take values in the sphere 𝕊 n .

Theorem A.4.

There exist ε > 0 , 0 < δ < 1 , α > 0 and C > 0 independent of u such that if u ∈ W 2 , 2 ⁢ ( B 1 , S n ) is a solution of

Δ 2 ⁢ u = Δ ⁢ ( V ⁢ ∇ ⁡ u ) + div ⁡ ( w ⁢ ∇ ⁡ u ) + ∇ ⁡ ω ⁢ ∇ ⁡ u + F ⁢ ∇ ⁡ u + f ,

where we have V ∈ W 1 , 2 ⁢ ( B 1 , M n + 1 ⊗ Λ 1 ⁢ R 4 ) , w ∈ L 2 ⁢ ( B 1 , M n + 1 ) , ω ∈ L 2 ⁢ ( B 1 , so n + 1 ) , F ∈ L 2 ⋅ W 1 , 2 ⁢ ( B 1 , M n + 1 ⊗ Λ 1 ⁢ R 4 ) and f ∈ L q ⁢ ( B 1 , R n + 1 ) with q > 1 , which satisfy (A.2) and

(A.4) ∥ Δ ⁢ u ∥ L 2 ⁢ ( B 1 ) ≤ ε ,

then we have u ∈ W loc 3 , 4 / 3 ⁢ ( B 1 , R n + 1 ) and

(A.5) ∥ ∇ 3 ⁡ u ∥ L 4 3 ⁢ ( B ⁢ ( p , ρ ) ) + ∥ ∇ 2 ⁡ u ∥ L 2 ⁢ ( B ⁢ ( p , ρ ) ) + ∥ ∇ ⁡ u ∥ L 4 ⁢ ( B ⁢ ( p , ρ ) )
  ≤ C ⁢ ρ α ⁢ ( ∥ Δ ⁢ u ∥ L 2 ⁢ ( B 1 ) + ∥ f ∥ L q ⁢ ( B 1 ) )

for all p ∈ B 1 4 and 0 ≤ ρ ≤ δ . Moreover, if f ∈ L 4 , 1 ⁢ ( B 1 ) (Lorentz space, see e.g. [20, 17]) then u ∈ W 3 , ∞ ⁢ ( B 1 16 , R n + 1 ) and for l = 1 , 2 , 3 we have

(A.6) | ∇ l ⁡ u | ⁢ ( 0 ) ≤ C l ⁢ ( ∥ Δ ⁢ u ∥ L 2 ⁢ ( B 1 ) + ∥ f ∥ L 4 , 1 ⁢ ( B 1 ) + ∥ f ∥ L 4 , 1 ⁢ ( B 1 ) 2 )

for some constant C l > 0 . In particular, by rescaling we have for x ∈ B 1 and l = 1 , 2 , 3 :

| ∇ l ⁡ u | ⁢ ( x ) ≤ C l ( 1 - | x | ) l ⁢ ( ∥ Δ ⁢ u ∥ L 2 ⁢ ( B 1 ) + ∥ f ∥ L 4 , 1 ⁢ ( B 1 ) + ∥ f ∥ L 4 , 1 ⁢ ( B 1 ) 2 ) .

Proof.

First of all, in order to apply Theorem A.3, we need ∥ ∇ 2 ⁡ u ∥ 2 2 + ∥ ∇ ⁡ u ∥ 4 4 to be small, and therefore we have to control ∥ ∇ 2 ⁡ u ∥ 2 up to reducing the size of the ball from the assumption (A.4). Since it is very important that all estimates are independent of the size of the ball, we give a proof on a ball of radius r ∈ ( 0 , 1 ] . Let u = ξ + η where ξ ∈ W 0 2 , 2 ⁢ ( B r ) and η ∈ W 2 , 2 ⁢ ( B r ) be such that

Δ ⁢ ξ = Δ ⁢ u

and

Δ ⁢ η = 0 .

Thanks to the standard L p theory (see e.g. [10, Theorem 4, Section 6.3]), we have

∥ ∇ 2 ⁡ ξ ∥ 2 ≤ C ⁢ ∥ Δ ⁢ u ∥ 2   on  ⁢ B r .

By classical theory of harmonic function, see e.g. [15, Corollary 1.37], we have

∫ B 3 ⁢ r / 4 | ∇ 2 ⁡ η | 2 ⁢ 𝑑 x ≤ C r 2 ⁢ ∫ B r | ∇ ⁡ η | 2 ⁢ 𝑑 x ,

where C > 0 is a universal constant. Hence, by the harmonicity of η and ξ = 0 on ∂ ⁡ B r (so that ∥ ∇ ⁡ u ∥ 2 2 = ∥ ∇ ⁡ ξ ∥ 2 2 + ∥ ∇ ⁡ η ∥ 2 2 on B r ), we have

∫ B 3 ⁢ r / 4 | ∇ 2 ⁡ η | 2 ⁢ 𝑑 x ≤ C r 2 ⁢ ∫ B r | ∇ ⁡ u | 2 ⁢ 𝑑 x ,

and thus

∫ B 3 ⁢ r / 4 | ∇ 2 ⁡ η | 2 ⁢ 𝑑 x ≤ C ⁢ ( ∫ B r | ∇ ⁡ u | 4 ⁢ 𝑑 x ) 1 2 .

Finally, using the fact that u takes values in 𝕊 n , we automatically have that

∥ ∇ ⁡ u ∥ 4 4 ≤ ∥ Δ ⁢ u ∥ 2 2 ,

which insures that (assuming ∥ Δ ⁢ u ∥ 2 is small)

(A.7) ∥ ∇ 2 ⁡ u ∥ 2 2 + ∥ ∇ ⁡ u ∥ 4 4 ≤ C ⁢ ∥ Δ ⁢ u ∥ 2   on  ⁢ B r .

Now assuming that ∥ ∇ 2 ⁡ u ∥ 2 2 + ∥ ∇ ⁡ u ∥ 4 4 is small on B 3 4 , thanks to (A.2) and (A.7), hypothesis of Theorem A.3 are satisfied on B 3 4 . Hence we can rewrite our equation as

Δ ⁢ ( A ⁢ Δ ⁢ u ) = div ⁡ ( K ) + A ⁢ f ,

where A ∈ L ∞ ∩ W 2 , 2 ⁢ ( B 3 8 , 𝒢 ⁢ l n + 1 ) and K ∈ L 2 ⋅ W 1 , 2 ⁢ ( B 3 8 ) ⊂ L 4 3 , 1 ⁢ ( B 3 8 ) satisfy

∥ A ∥ W 2 , 2 + d ⁢ ( A , 𝒮 ⁢ 𝒪 n + 1 ) ≤ C ⁢ ( ∥ ∇ 2 ⁡ u ∥ 2 + ∥ ∇ ⁡ u ∥ 4 )

and

∥ K ∥ L 4 3 , 1 ≤ C ⁢ ( ∥ ∇ 2 ⁡ u ∥ 2 2 + ∥ ∇ ⁡ u ∥ 4 2 ) ,

where C > 0 is independent of u.

Now let p ∈ B 1 4 and 0 < ρ < 1 8 so that B ρ ⁢ ( p ) ⊂ B 3 8 . Use the Hodge decomposition we decompose A ⁢ Δ ⁢ u on B ρ ⁢ ( p ) as

A ⁢ Δ ⁢ u = R + S ,

where R ∈ W 0 1 , 2 ⁢ ( B ρ ⁢ ( p ) ) and S ∈ W 1 , 2 ⁢ ( B ρ ⁢ ( p ) ) such that R satisfies

Δ ⁢ R = div ⁡ ( K ) + A ⁢ f

and S satisfies

Δ ⁢ S = 0

on B ρ ⁢ ( p ) . Thanks to the standard L p -theory and Sobolev embeddings, on B ρ ⁢ ( p ) we get (using that q > 1 )

(A.8) ∥ R ∥ 2 ≤ C ⁢ ( ∥ K ∥ 4 / 3 + ∥ f ∥ q + 1 2 )
≤ C ⁢ ( ε 1 4 ⁢ ( ∥ ∇ 2 ⁡ u ∥ 2 + 1 ρ ⁢ ∥ ∇ ⁡ u ∥ 2 ) + ρ q - 1 q ⁢ ( q + 1 ) ⁢ ∥ f ∥ q ) ,

where C > 0 is independent of u. Now using the fact that S is harmonic, thanks to Lemma A.5 we have that γ ↦ 1 ( γ ⁢ ρ ) 4 ⁢ ∫ B γ ⁢ ρ ⁢ ( p ) | S | 2 ⁢ 𝑑 x is an increasing function and hence for all γ ∈ ( 0 , 1 ) we have

(A.9) ∫ B γ ⁢ ρ ⁢ ( p ) | S | 2 ⁢ 𝑑 x ≤ γ 4 ⁢ ∫ B ρ ⁢ ( p ) | S | 2 ⁢ 𝑑 x .

We then decompose u as u = E + F , where E ∈ W 0 1 , 4 ⁢ ( B ρ ⁢ ( p ) ) and F ∈ W 1 , 4 ⁢ ( B ρ ⁢ ( p ) ) satisfy

Δ ⁢ E = A - 1 ⁢ ( R + S ) ⁢  on  ⁢ B ρ ⁢ ( p )

and

Δ ⁢ F = 0 ⁢  on  ⁢ B ρ ⁢ ( p ) .

Thanks again to the standard L p -theory and Sobolev embeddings, on B ρ ⁢ ( p ) we get

(A.10) 1 ρ ⁢ ∥ ∇ ⁡ E ∥ 2 ≤ C ⁢ ( ∥ R ∥ 2 + ∥ S ∥ 2 ) ,

where C > 0 is independent of u. Note that ∥ ∇ ⁡ u ∥ 2 = ∥ ∇ ⁡ E ∥ 2 + ∥ ∇ ⁡ F ∥ 2 on B ρ ⁢ ( p ) .

Now the function γ ↦ 1 ( γ ⁢ ρ ) 4 ⁢ ∫ B γ ⁢ ρ ⁢ ( p ) | ∇ ⁡ F | 2 ⁢ 𝑑 x is increasing since F is harmonic and we have again, for all γ ∈ ( 0 , 1 ) ,

(A.11) 1 ( γ ⁢ ρ ) 2 ⁢ ∫ B γ ⁢ ρ ⁢ ( p ) | ∇ ⁡ F | 2 ⁢ 𝑑 x ≤ γ 2 ρ 2 ⁢ ∫ B ρ ⁢ ( p ) | ∇ ⁡ F | 2 ⁢ 𝑑 x .

Then, thanks to (A.8), (A.9), (A.10) and (A.11), for γ and ε small enough (with respect to some constant independent of u), we have

∫ B γ ⁢ ρ ⁢ ( p ) ( | ∇ 2 ⁡ u | 2 + | ∇ ⁡ u | 2 ( γ ⁢ ρ ) 2 ) ⁢ 𝑑 x ≤ 1 2 ⁢ ∫ B ρ ⁢ ( p ) ( | ∇ 2 ⁡ u | 2 + | ∇ ⁡ u | 2 ρ 2 ) ⁢ 𝑑 x + C ⁢ ( γ ⁢ ρ ) 2 ⁢ ( q - 1 ) q ⁢ ( q + 1 ) ⁢ ∥ f ∥ q 2 .

Here, we have used that the L 2 -norm of Hessian is controlled by the L 2 -norms of the Laplacian and the gradient, up to reducing the size of the ball, see (A.7). Iterating this inequality gives the following Morrey-type estimate: there exists 0 < δ < 1 8 , α > 0 and C > 0 independent of u such that

(A.12) sup p ∈ B 1 4 , 0 < ρ < δ ⁡ ρ - α ⁢ ( ∫ B ρ ⁢ ( p ) ( | ∇ 2 ⁡ u | 2 + 1 ρ 2 ⁢ | ∇ ⁡ u | 2 ) ⁢ 𝑑 x )
≤ C ⁢ ( ∥ Δ ⁢ u ∥ L 2 ⁢ ( B 1 ) 2 + ∥ f ∥ L q ⁢ ( B 1 ) 2 ) .

Then, for any p ∈ B 1 4 and 0 < ρ < δ we have

∥ K ∥ L 4 3 ⁢ ( B ⁢ ( p , ρ ) ) ≤ C ⁢ ρ α ⁢ ( ∥ Δ ⁢ u ∥ L 2 ⁢ ( B 1 ) 2 + ∥ f ∥ L q ⁢ ( B 1 ) 2 ) .

Setting A ⁢ Δ ⁢ u = R + S on B ⁢ ( p , ρ ′ ) with ρ ′ ∈ ( 3 ⁢ ρ 4 , ρ ) as before (S is harmonic), where ρ ′ will be fixed later, we get

(A.13) ∥ ∇ ⁡ R ∥ L 4 3 ⁢ ( B ⁢ ( p , ρ 2 ) ) ≤ C ⁢ ρ α ⁢ ( ∥ Δ ⁢ u ∥ L 2 ⁢ ( B 1 ) 2 + ∥ f ∥ L q ⁢ ( B 1 ) 2 ) .

Using a Green formula, we get for all y ∈ B ⁢ ( p , ρ 2 ) that

| ∇ ⁡ S ⁢ ( y ) | ≤ C ρ 4 ⁢ ∫ ∂ ⁡ B ⁢ ( p , ρ ′ ) | A ⁢ Δ ⁢ u | ⁢ 𝑑 x
≤ C ⁢ ρ 3 2 ρ 4 ⁢ ( ∫ ∂ ⁡ B ⁢ ( p , ρ ′ ) | A ⁢ Δ ⁢ u | 2 ⁢ 𝑑 x ) 1 2
≤ C ⁢ ρ 3 2 ρ 4 ⁢ ( 4 ρ ⁢ ∫ B ⁢ ( p , ρ ) ∖ B ⁢ ( p , 3 ⁢ ρ 4 ) | A ⁢ Δ ⁢ u | 2 ⁢ 𝑑 x ) 1 2 ,

the last inequality is obtained thanks to the mean value theorem, by choosing ρ ′ correctly. Thanks to (A.3) and (A.12), for any p ∈ B 1 4 and all y ∈ B ⁢ ( p , ρ 2 ) we get

| ∇ ⁡ S ⁢ ( y ) | ≤ C ⁢ ρ α 2 - 3 ⁢ ( ∥ Δ ⁢ u ∥ L 2 ⁢ ( B 1 ) + ∥ f ∥ L q ⁢ ( B 1 ) )

and

(A.14) ∥ ∇ ⁡ S ∥ L 4 3 ⁢ ( B ⁢ ( p , ρ 2 ) ) ≤ C ⁢ ρ α 2 ⁢ ( ∥ Δ ⁢ u ∥ L 2 ⁢ ( B 1 ) + ∥ f ∥ L q ⁢ ( B 1 ) ) .

Thanks to (A.13) and (A.14), we get for any p ∈ B 1 4 and all 0 < ρ < δ 4 ,

(A.15) ( ∫ B ( p , ρ 2 ) ) | ∇ ⁡ ( A ⁢ Δ ⁢ u ) | 4 3 ⁢ 𝑑 x ) 3 4 ≤ C ⁢ ρ α 2 ⁢ ( ∥ Δ ⁢ u ∥ L 2 ⁢ ( B 1 ) + ∥ f ∥ L q ⁢ ( B 1 ) ) .

Finally, thanks to (A.12) and (A.15), we get (A.5).

Then we can bootstrap those estimates so that there exists β > 0 such that

sup p ∈ B 1 8 ,  0 < ρ < δ 8 ⁡ ρ - β ⁢ ∫ B ρ ⁢ ( p ) | Δ 2 ⁢ u | ⁢ 𝑑 x ≤ C ⁢ ( ∥ Δ ⁢ u ∥ L 2 ⁢ ( B 1 ) 2 + ∥ f ∥ L q ⁢ ( B 1 ) 2 ) .

Now suppose f ∈ L 4 , 1 ⁢ ( B 1 ) instead of L q ⁢ ( B 1 ) for some q > 1 and the above estimates are still valid by replacing ∥ f ∥ L q ⁢ ( B 1 ) with ∥ f ∥ L 4 , 1 ⁢ ( B 1 ) . Then a classical Green formula gives, for all p ∈ B 1 8 ,

| ∇ 3 ⁡ u | ⁢ ( p ) ≤ C ⁢ 1 | x - p | 3 ∗ χ B 1 / 4 ⁢ | Δ 2 ⁢ u | + C ⁢ ( ∥ Δ ⁢ u ∥ L 2 ⁢ ( B 1 ) + ∥ f ∥ L 4 , 1 ⁢ ( B 1 ) ) , | ∇ 2 ⁡ u | ⁢ ( p ) ≤ C ⁢ 1 | x - p | 2 ∗ χ B 1 4 ⁢ | Δ 2 ⁢ u | + C ⁢ ( ∥ Δ ⁢ u ∥ L 2 ⁢ ( B 1 ) + ∥ f ∥ L 4 , 1 ⁢ ( B 1 ) ) , | ∇ ⁡ u | ⁢ ( p ) ≤ C ⁢ 1 | x - p | ∗ χ B 1 4 ⁢ | Δ 2 ⁢ u | + C ⁢ ( ∥ Δ ⁢ u ∥ L 2 ⁢ ( B 1 ) + ∥ f ∥ L 4 , 1 ⁢ ( B 1 ) ) ,

where χ B 1 4 is the characteristic function of the ball B 1 4 . We use the Green formula on a cut-off of u, the remaining terms are easily control thanks to (A.5). Together with injections proved by Adams in [1], see also [12, Exercise 6.1.6], the latter shows that

∥ ∇ 3 ⁡ u ∥ L r ⁢ ( B 1 8 ) + ∥ ∇ 2 ⁡ u ∥ L r ⁢ ( B 1 8 ) + ∥ ∇ ⁡ u ∥ L r ⁢ ( B 1 8 )
≤ C ⁢ ( ∥ Δ ⁢ u ∥ L 2 ⁢ ( B 1 ) + ∥ f ∥ L 4 , 1 ⁢ ( B 1 ) + ∥ f ∥ L 4 , 1 ⁢ ( B 1 ) 2 )

for some r > 4 3 . Then bootstrapping this estimate, we get

∥ ∇ 3 ⁡ u ∥ L q ¯ ⁢ ( B 1 16 ) + ∥ ∇ 2 ⁡ u ∥ L p ¯ ⁢ ( B 1 16 ) + ∥ ∇ ⁡ u ∥ L p ¯ ⁢ ( B 1 16 )
≤ C ⁢ ( ∥ Δ ⁢ u ∥ L 2 ⁢ ( B 1 ) + ∥ f ∥ L 4 , 1 ⁢ ( B 1 ) + ∥ f ∥ L 4 , 1 ⁢ ( B 1 ) 2 ) ,

where q ¯ is the limiting exponent of the bootstrapping given by the Sobolev injection of W 1 , q into L q ¯ . Indeed, thanks to (A.2), the only limiting term for the bootstrap is the regularity of f. But thanks to the embedding of W 1 , ( 4 , 1 ) into L ∞ , see e.g. [22] and [43], we can conclude the proof of the theorem. ∎

Lemma A.5.

Let v be a harmonic function on B 1 . For every point p in B 1 , the function

ρ ↦ 1 ρ 4 ⁢ ∫ B ⁢ ( p , ρ ) | v | 2 ⁢ 𝑑 x

is increasing.

Proof.

We have

(A.16) d d ⁢ ρ ⁢ [ 1 ρ 4 ⁢ ∫ B ⁢ ( p , ρ ) | v | 2 ⁢ 𝑑 x ] = - 4 ρ 5 ⁢ ∫ B ⁢ ( p , ρ ) | v | 2 ⁢ 𝑑 x + 1 ρ 4 ⁢ ∫ ∂ ⁡ B ⁢ ( p , ρ ) | v | 2 ⁢ 𝑑 σ .

Let ( ϕ k l ) l , k be an L 2 -basis of eigenfunctions of the Laplacian on 𝕊 3 . In particular,

Δ ⁢ ϕ k l = - l ⁢ ( l + 2 ) ⁢ ϕ k l .

We have

v ⁢ ( ρ , θ ) = ∑ l = 0 + ∞ ∑ k = 1 N l a k l ⁢ ϕ k l   on  ⁢ ∂ ⁡ B ⁢ ( p , ρ ) ,

where N l is the dimension of the eigenspace corresponding to - l ⁢ ( l + 2 ) , see [13]. Hence

v ⁢ ( r , θ ) = ∑ l = 0 + ∞ ∑ k = 1 N l a k l ⁢ ( r ρ ) l ⁢ ϕ k l   on  ⁢ B ⁢ ( p , ρ ) .

Then

(A.17) ∫ ∂ ⁡ B ⁢ ( p , ρ ) | v | 2 ⁢ 𝑑 x = ∑ l = 0 + ∞ ∑ k = 1 N l | a k l | 2 ⁢ ρ 3

and

(A.18) ∫ B ⁢ ( p , ρ ) | v | 2 ⁢ 𝑑 x = ∑ l = 0 + ∞ ∑ k = 1 N l | a k l | 2 2 ⁢ l + 4 ⁢ ρ 4 .

Finally, putting (A.16), (A.17) and (A.18) together, we get the desired result. ∎

B Further remarks and open questions

By the proof of Theorem A.4 we know that ∥ Δ 2 ⁢ u ∥ L loc 1 ⁢ ( B 1 ) is small for a W 2 , 2 weakly intrinsic or extrinsic bi-harmonic map u defined on B 1 with small bi-energy. Analogously, in [27, Theorem A.4] for weakly harmonic maps, | Δ ⁢ u | ≃ | ∇ ⁡ u | 2 is estimated to be in the local Hardy space h 1 ⊊ L 1 for weakly harmonic maps with small Dirichlet energy on the 2-disk. In this case, the improved global Hardy estimate turns out to be equivalent to the use of the (first-order) Hardy inequality plus the ε-regularity | ∇ ⁡ u | ≤ C ⁢ ε 0 1 - | x | for weakly harmonic maps on the 2-disk. It is worth remarking that such improved global estimate is a typical compensation phenomenon for the special Jacobian structure of the harmonic map equation

- Δ ⁢ u = A ⁢ ( u ) ⁢ ( ∇ ⁡ u , ∇ ⁡ u ) = Ω ⋅ ∇ ⁡ u ,

where Ω is anti-symmetric. Given the special structure of the intrinsic or extrinsic bi-harmonic map equation, it is quite natural to conjecture that | Δ 2 ⁢ u | is in L 1 ⁢ ( B 1 ) (with small L 1 -norm), but as far as we know this has not yet been proved. We leave it as an interesting open question. If this was true, given the ε-regularity Theorem A.4 and the following theorem (Theorem B.1), then one can bypass the use of the Hardy inequality (Theorem 2.1) in the proof of the energy convexity results (Theorems 1.4, 2.2 and 1.9), which we will explain below. In a forthcoming paper, we will explore more on this interesting relations between the Hardy inequality, ε-regularity, compensation phenomenon of the structure of the bi-harmonic map equations and the global integrability of the solution u.

Theorem B.1.

Let f ≥ 0 satisfy

f ⁢ ( x ) ≤ C 0 ( 1 - | x | ) 4   for a.e. ⁢ x ∈ B 1   𝑎𝑛𝑑   ∥ f ∥ 1 ≤ C 0

for some constant C 0 > 0 . Then there exists a function ψ ∈ L ∞ ∩ W 2 , 2 ⁢ ( B 1 ) solving the boundary value problem

{ Δ 2 ⁢ ψ = f in  ⁢ B 1 , ψ = ∂ ν ⁡ ψ = 0 on  ⁢ ∂ ⁡ B 1 .

Moreover, there exists a constant C > 0 such that

(B.1) ∥ Δ ⁢ ψ ∥ 2 + ∥ ∇ ⁡ ψ ∥ 4 + ∥ ψ ∥ ∞ ≤ C ⁢ C 0 .

Proof.

The idea of the proof follows [40, Proposition 1.68]. Since the Green’s function of Δ 2 on B 1 with the clamped plate boundary values is given explicitly by

G ⁢ ( x , y ) = c ⁢ ( ln ⁡ | x - y | - ln ⁡ ( | x | x | - | x | ⁢ y | ) - | x - y | 2 2 ⁢ | x | x | - | ⁢ x ⁢ | y | 2 + 1 2 )

for some normalizing constant c < 0 (see [3] or [14, Lemma 2.1]), we can write

(B.2) ψ ⁢ ( x ) = c ⁢ ∫ B 1 f ⁢ ( y ) ⁢ ( ln ⁡ | x - y | - ln ⁡ ( | x | x | - | x | ⁢ y | ) - | x - y | 2 2 ⁢ | x | x | - | ⁢ x ⁢ | y | 2 + 1 2 ) ⁢ 𝑑 y .

Let θ ∈ C 0 ∞ ⁢ ( B 1 ) be a smooth bump function such that 0 ≤ θ ≤ 1 , θ = 1 in B 1 16 and spt ⁢ ( θ ) ⊂ B 1 8 . For x ∈ B 1 we define

(B.3) l x ⁢ ( y ) := ∑ j = 0 ∞ θ ⁢ ( 2 j ⁢ ( 1 - | x | ) - 1 ⁢ ( x - y ) )   for  ⁢ y ∈ B 1 .

We claim that for any x , y ∈ B 1 ,

(B.4) - 20 ln 2 ≤ ln | x - y | - ln ( | x | x | - | x | y | ) - | x - y | 2 2 ⁢ | x | x | - | ⁢ x ⁢ | y | 2 + 1 2 + l x ( y ) ln 2
≤ 20 ⁢ ln ⁡ 2 ,

To see this, it is clear that for x , y ∈ B 1 such that

(B.5) 2 - k ≤ | x - y | ≤ 2 - k + 1 , k ∈ ℕ 0 ,

we have

(B.6) - k ⁢ ln ⁡ 2 ≤ ln ⁡ | x - y | ≤ ( - k + 1 ) ⁢ ln ⁡ 2 .

Now note that

1 - | x | - | x - y | ≤ 1 - | x | + | x | - | y | = 1 - | y | ≤ 1 - | x | + | x - y | ,

and therefore for x ∈ B 1 - 2 - i - 1 ∖ B 1 - 2 - i , i.e., 1 - | x | ∈ [ 2 - i - 1 , 2 - i ] , i ∈ ℕ 0 (with B ¯ 0 = ∅ ) and any y ∈ B 1 satisfying (B.5), we have

1 - | y | ∈ { [ 2 - i - 1 - 2 - k + 1 , 2 - i + 2 - k + 1 ] if  ⁢ k ≥ i + 4 , [ 0 , 2 - i + 2 - k + 1 ] if  ⁢ k ≤ i + 3 .

We also have

0 ≤ ( 1 - | x | ) ⁢ ( 1 - | y | ) ≤ ( 1 - | x | 2 ) ⁢ ( 1 - | y | 2 )
= | x | x | - | ⁢ x ⁢ | y | 2 - | x - y | 2 ≤ 2 2 ⁢ ( 1 - | x | ) ⁢ ( 1 - | y | ) ,

and thus

| x | x | - | x | y | 2 - | x - y | 2 ∈ { [ 2 - 2 ⁢ i - 2 - 2 - i - k , 2 - 2 ⁢ i + 2 + 2 - i - k + 3 ] if  ⁢ k ≥ i + 4 , [ 0 , 2 - 2 ⁢ i + 2 + 2 - i - k + 3 ] if  ⁢ k ≤ i + 3 .

Combining this with (B.5), we get

| x | x | - | x | y | 2 ∈ { [ 2 - 2 ⁢ i - 2 - 2 - i - k + 2 - 2 ⁢ k , 2 - 2 ⁢ i + 2 + 2 - i - k + 3 + 2 - 2 ⁢ k + 2 ] if  ⁢ k ≥ i + 4 , [ 2 - 2 ⁢ k , 2 - 2 ⁢ i + 2 + 2 - i - k + 3 + 2 - 2 ⁢ k + 2 ] if  ⁢ k ≤ i + 3 .

Now using the facts that for k ≥ i + 4 we have

2 - 2 ⁢ i - 2 - 2 - i - k + 2 - 2 ⁢ k ≥ 2 - 2 ⁢ i - 4   and   2 - 2 ⁢ i + 2 + 2 - i - k + 3 + 2 - 2 ⁢ k + 2 ≤ 2 - 2 ⁢ i + 4

and for k ≤ i + 3 we have

2 - 2 ⁢ i + 2 + 2 - i - k + 3 + 2 - 2 ⁢ k + 2 ≤ 2 - 2 ⁢ k + 10 ,

we arrive at

| x | x | - | x | y | 2 ∈ { [ 2 - 2 ⁢ i - 4 , 2 - 2 ⁢ i + 4 ] if  ⁢ k ≥ i + 4 , [ 2 - 2 ⁢ k , 2 - 2 ⁢ k + 10 ] if  ⁢ k ≤ i + 3 ,

and hence

(B.7) - ln | x | x | - | x | y | ∈ { [ ( i - 2 ) ⁢ ln ⁡ 2 , ( i + 2 ) ⁢ ln ⁡ 2 ] if  ⁢ k ≥ i + 4 , [ ( k - 5 ) ⁢ ln ⁡ 2 , k ⁢ ln ⁡ 2 ] if  ⁢ k ≤ i + 3 .

Combining (B.6) and (B.7), we get

(B.8) ln | x - y | - ln ( | x | x | - | x | y | )
  ∈ { [ ( - k + i - 2 ) ⁢ ln ⁡ 2 , ( - k + i + 3 ) ⁢ ln ⁡ 2 ] if  ⁢ k ≥ i + 4 , [ - 5 ⁢ ln ⁡ 2 , ln ⁡ 2 ]   ( in fact,  ⁢ [ - 5 ⁢ ln ⁡ 2 , 0 ] ) if  ⁢ k ≤ i + 3 ,

and

(B.9) | x - y | 2 2 ⁢ | x | x | - | ⁢ x ⁢ | y | 2 ∈ { [ 0 , 1 4 ] if  ⁢ k ≥ i + 4 , [ 2 - 6 , 1 ] if  ⁢ k ≤ i + 3 ,

for any x ∈ B 1 - 2 - i - 1 ∖ B 1 - 2 - i , i ≥ 0 , and any y ∈ B 1 satisfying (B.5) for some k ≥ 0 .

Now for any x ∈ B 1 - 2 - i - 1 ∖ B 1 - 2 - i , i ≥ 0 , and any y ∈ B 1 satisfying estimate (B.5), since 0 ≤ θ ≤ 1 , θ = 1 in B 1 16 and spt ⁢ ( θ ) ⊂ B 1 8 , we get that for any j ≥ 0 ,

θ ⁢ ( 2 j ⁢ ( 1 - | x | ) - 1 ⁢ ( x - y ) ) = 0   for  ⁢ | x - y | ≥ 2 - j - 3 ⁢ ( 1 - | x | ) ∈ [ 2 - j - i - 4 , 2 - j - i - 3 ]

and

θ ⁢ ( 2 j ⁢ ( 1 - | x | ) - 1 ⁢ ( x - y ) ) = 1   for  ⁢ | x - y | ≤ 2 - j - 4 ⁢ ( 1 - | x | ) ∈ [ 2 - j - i - 5 , 2 - j - i - 4 ] .

Therefore (combining with (B.5)),

(B.10) θ ⁢ ( 2 j ⁢ ( 1 - | x | ) - 1 ⁢ ( x - y ) ) = 0   for  ⁢ j ≥ k - i - 3

and

(B.11) θ ( 2 j ( 1 - | x | ) - 1 ( x - y ) ) = 1   if  k - 1 ≥ j + i + 5   ( i.e.  j ≤ k - i - 6 ) .

Hence for any x ∈ B 1 - 2 - i - 1 ∖ B 1 - 2 - i , i ≥ 0 , and any y ∈ B 1 such that

2 - k ≤ | x - y | ≤ 2 - k + 1

for some k = 0 , 1 , 2 , … , (B.3), (B.10) and (B.11) imply

(B.12) { k - i - 10 ≤ l x ⁢ ( y ) ≤ k - i + 10 if  ⁢ k ≥ i + 4 , l x ⁢ ( y ) = 0 if  ⁢ k ≤ i + 3 .

Combining (B.8), (B.9) and (B.12) gives (B.4).

Therefore, in order to obtain the L ∞ -bound of ψ on B 1 as in (B.1), it suffices to bound ∫ B 1 f ⁢ ( y ) ⁢ l x ⁢ ( y ) ⁢ 𝑑 y since (B.2) and (B.4) hold. Therefore, using the facts that f ≥ 0 , 0 ≤ θ ≤ 1 and spt ⁡ ( θ ) ⊂ B 1 8 , for any x ∈ B 1 we have

(B.13) | ∫ B 1 f ⁢ ( y ) ⁢ l x ⁢ ( y ) ⁢ 𝑑 y | ≤ ∑ j = 0 ∞ ∫ B 1 f ⁢ ( y ) ⁢ θ ⁢ ( 2 j ⁢ ( 1 - | x | ) - 1 ⁢ ( x - y ) ) ⁢ 𝑑 y
= ∑ j = 0 ∞ ∫ B 2 - j - 3 ⁢ ( 1 - | x | ) ⁢ ( x ) f ⁢ ( y ) ⁢ θ ⁢ ( 2 j ⁢ ( 1 - | x | ) - 1 ⁢ ( x - y ) ) ⁢ 𝑑 y
≤ ∑ j = 0 ∞ ∫ B 2 - j - 3 ⁢ ( 1 - | x | ) ⁢ ( x ) C 0 ( 1 - | y | ) 4 ⁢ 𝑑 y
≤ ∑ j = 0 ∞ ∫ B 2 - j - 3 ⁢ ( 1 - | x | ) ⁢ ( x ) C 0 ( 7 8 ) 4 ⁢ ( 1 - | x | ) 4 ⁢ 𝑑 y
≤ C 0 ⁢ ( 8 7 ) 4 ⁢ π 2 2 ⁢ ∑ j = 0 ∞ 2 - 4 ⁢ j - 12
≤ C ⁢ C 0 .

Combining (B.2), (B.4), (B.13) and the fact that ∥ f ∥ 1 ≤ C 0 yields

| ψ ⁢ ( x ) | ≤ C ⁢ C 0 .

This gives the desired L ∞ -bound of ψ on B 1 . The L 2 -estimate for Δ ⁢ ψ and L 4 -estimate for ∇ ⁡ ψ simply follows from an integration by parts argument. ∎

Now let us explain how we can bypass the Hardy inequality (Theorem 2.1) in the proof of the energy convexity (Theorems 1.4, 2.2 and 1.9) using Theorem B.1 and the ε-regularity Theorem A.4, assuming that the L 1 -norm of Δ 2 ⁢ u is small for the intrinsic or extrinsic bi-harmonic maps. This is a quite natural assumption for most of the applications, for example, this is valid if u ∈ W 3 , 4 3 ⁢ ( B 1 ) and ∥ ∇ 2 ⁡ u ∥ L 2 is small or if the bi-harmonic map is defined on a larger set containing B 1 .

Lemma B.2.

There exist constants ε 0 > 0 , C > 0 such that if u ∈ W 2 , 2 ⁢ ( B 1 , S n ) is a weakly intrinsic or extrinsic bi-harmonic map with

∫ B 1 | Δ ⁢ u | 2 ⁢ 𝑑 x ≤ ε 0   𝑎𝑛𝑑   ∫ B 1 | Δ 2 ⁢ u | ⁢ 𝑑 x ≤ ε 0 ,

then there exists a solution ϕ of

(B.14) { Δ 2 ⁢ ϕ = q ⁢ ( u ) on  ⁢ B 1 , ϕ = ∂ ν ⁡ ϕ = 0 on  ⁢ ∂ ⁡ B 1 ,

such that

(B.15) ∥ Δ ⁢ ϕ ∥ 2 + ∥ ∇ ⁡ ϕ ∥ 4 + ∥ ϕ ∥ ∞ ≤ C ⁢ ε 0 .

Here q ⁢ ( u ) = | ∇ ⁡ u | 4 , | ∇ ⁡ u | 2 ⁢ | Δ ⁢ u | , | ∇ 2 ⁡ u | 2 or | ∇ ⁡ u | ⁢ | ∇ ⁡ Δ ⁢ u | .

Proof.

Note that by the ε-regularity Theorem A.4 (with f ≡ 0 ), we have

| q ⁢ ( u ) | ⁢ ( x ) ≤ C ⁢ ∥ Δ ⁢ u ∥ 2 2 ⁢ ( 1 - | x | ) - 4 ≤ C ⁢ ε 0 ⁢ ( 1 - | x | ) - 4

for a.e. x ∈ B 1 . We can then apply Theorem B.1 (with C 0 = C ⁢ ε 0 ) to conclude the proof of this lemma. ∎

Lemma B.3.

There exists a constant ε 0 > 0 such that if u , v ∈ W 2 , 2 ⁢ ( B 1 , S n ) with u | ∂ ⁡ B 1 = v | ∂ ⁡ B 1 , ∂ ν ⁡ u | ∂ ⁡ B 1 = ∂ ν ⁡ v | ∂ ⁡ B 1 ,

∫ B 1 | Δ ⁢ u | 2 ⁢ 𝑑 x ≤ ε 0   𝑎𝑛𝑑   ∫ B 1 | Δ 2 ⁢ u | ⁢ 𝑑 x ≤ ε 0 ,

and u is a weakly intrinsic or extrinsic bi-harmonic map, then there exists C > 0 such that

∫ B 1 | v - u | 2 ⁢ q ⁢ ( u ) ⁢ 𝑑 x ≤ C ⁢ ε 0 ⁢ ∫ B 1 | Δ ⁢ ( v - u ) | 2 ⁢ 𝑑 x .

Here q ⁢ ( u ) = | ∇ ⁡ u | 4 , | ∇ ⁡ u | 2 ⁢ | Δ ⁢ u | , | ∇ 2 ⁡ u | 2 or | ∇ ⁡ u | ⁢ | ∇ ⁡ Δ ⁢ u | .

Remark B.4.

This estimate replaces the direct use of the Hardy inequality (Theorem 2.1) in the proofs of Theorems 1.4, 2.2 and 1.9 and then yields new proofs assuming ∥ Δ 2 ⁢ u ∥ L 1 ⁢ ( B 1 ) is small, but more importantly this shows that the compensation phenomena obtained in Lemma B.2 using the ϵ-regularity (Theorem A.4) is almost equivalent to the Hardy inequality.

Proof.

Let ϕ be a solution of (B.14), then

(B.16) ∫ B 1 | v - u | 2 ⁢ q ⁢ ( u ) ⁢ 𝑑 x
  = ∫ B 1 | v - u | 2 ⁢ Δ 2 ⁢ ϕ ⁢ 𝑑 x = ∫ B 1 Δ ⁢ ( | v - u | 2 ) ⁢ Δ ⁢ ϕ ⁢ 𝑑 x
  = 2 ⁢ ∫ B 1 〈 Δ ⁢ ( v - u ) , v - u 〉 ⁢ Δ ⁢ ϕ ⁢ 𝑑 x + 2 ⁢ ∫ B 1 | ∇ ⁡ ( v - u ) | 2 ⁢ Δ ⁢ ϕ ⁢ 𝑑 x
  ≤ 2 ( ( ∫ B 1 | Δ ( v - u ) | 2 d x ) 1 2 ( ∫ B 1 | ( v - u ) | 2 | Δ ϕ | 2 d x ) 1 2
      + ∥ ∇ ( v - u ) ∥ 4 2 ∥ Δ ϕ ∥ 2 ) .

Remarking that we have (2.3), then Lemma B.2 permits to control the second term in (B.16) as desired. Now for the first term in (B.16), we have

(B.17) ∫ B 1 | v - u | 2 ⁢ | Δ ⁢ ϕ | 2 ⁢ 𝑑 x
  = - 2 ⁢ ∫ B 1 〈 ∇ ⁡ ( v - u ) , ( v - u ) ⁢ ∇ ⁡ ϕ 〉 ⁢ Δ ⁢ ϕ ⁢ 𝑑 x - ∫ B 1 | v - u | 2 ⁢ 〈 ∇ ⁡ ϕ , ∇ ⁡ Δ ⁢ ϕ 〉 ⁢ 𝑑 x
  = - 2 ⁢ ∫ B 1 〈 ∇ ⁡ ( v - u ) , ( v - u ) ⁢ ∇ ⁡ ϕ 〉 ⁢ Δ ⁢ ϕ ⁢ 𝑑 x
      + 2 ⁢ ∫ B 1 〈 ∇ ⁡ ( v - u ) , ( v - u ) ⁢ ϕ ⁢ ∇ ⁡ Δ ⁢ ϕ 〉 ⁢ 𝑑 x + ∫ B 1 | v - u | 2 ⁢ ϕ ⁢ Δ 2 ⁢ ϕ ⁢ 𝑑 x
  = - 4 ⁢ ∫ B 1 〈 ∇ ⁡ ( v - u ) , ( v - u ) ⁢ ∇ ⁡ ϕ 〉 ⁢ Δ ⁢ ϕ ⁢ 𝑑 x - 2 ⁢ ∫ B 1 | ∇ ⁡ ( v - u ) | 2 ⁢ ϕ ⁢ Δ ⁢ ϕ ⁢ 𝑑 x
      - 2 ⁢ ∫ B 1 〈 Δ ⁢ ( v - u ) , v - u 〉 ⁢ ϕ ⁢ Δ ⁢ ϕ ⁢ 𝑑 x + ∫ B 1 | v - u | 2 ⁢ ϕ ⁢ Δ 2 ⁢ ϕ ⁢ 𝑑 x
  ≤ C ( ∥ ∇ ϕ ∥ 4 ∥ ∇ ( v - u ) ∥ 4 ( ∫ B 1 | v - u | 2 | Δ ϕ | 2 d x ) 1 2
      + ∥ ϕ ∥ ∞ ⁢ ∥ Δ ⁢ ϕ ∥ 2 ⁢ ( ∫ B 1 |