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

Proposition A.1.

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

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

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

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

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

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

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

and

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

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

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

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

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

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

and

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

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

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

and thus

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

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

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

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

and

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

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

and S satisfies

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

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

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

and

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

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 ) ,

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

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

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

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

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

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

and

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

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

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 ,

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

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

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

is increasing.

Proof.

We have

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

We have

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

Then

and

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

Theorem B.1.

Let f ≥ 0 satisfy

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

Moreover, there exists a constant C > 0 such that

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

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

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

We claim that for any x , y ∈ B 1 ,

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

we have

Now note that

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

We also have

and thus

Combining this with (B.5), we get

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

and for k ≤ i + 3 we have

we arrive at

and hence

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

and

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 ,

and

Therefore (combining with (B.5)),

and

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

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

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

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

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. ∎