# Abstract

In this paper, we show an energy convexity and thus uniqueness for weakly intrinsic bi-harmonic maps from the unit 4-ball

## 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

## Proposition A.1.

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

*where we have
*

*almost everywhere, where
*

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

*There exist constants
*

*then there exist
*

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
*

*where we have
*

*then there exist
*

*and*

A first consequence of Theorem A.3, is the ε-regularity for approximate intrinsic and extrinsic bi-harmonic maps into

## Theorem A.4.

*There exist
*

*u*such that if

*where we have
*

*then we have
*

*for all
*

*for some constant
*

## Proof.

First of all, in order to apply Theorem A.3, we need

and

Thanks to the standard

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

where

and thus

Finally, using the fact that *u* takes values in

which insures that (assuming

Now assuming that

where

and

where
*u*.

Now let

where
*R* satisfies

and *S* satisfies

on

where
*u*. Now using the fact that *S* is harmonic, thanks to Lemma A.5 we have that

We then decompose *u* as

and

Thanks again to the standard

where
*u*. Note that

Now the function
*F* is harmonic and we have again, for all

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
*u* such that

Then, for any

Setting
*S* is harmonic), where

Using a Green formula, we get for all

the last inequality is obtained thanks to the mean value theorem, by choosing

and

Thanks to (A.13) and (A.14), we get for any

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

Then we can bootstrap those estimates so that there exists

Now suppose

where
*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

where
*f*. But thanks to the embedding of

## Lemma A.5.

*Let v be a harmonic function on
*

*is increasing.*

## Proof.

We have

Let

We have

where

Then

and

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
*u* defined on

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
*u*.

## Theorem B.1.

*Let
*

*for some constant
*

*Moreover, there exists a constant
*

## Proof.

The idea of the proof follows [40, Proposition 1.68]. Since the Green’s function of

for some normalizing constant

Let

We claim that for any

To see this, it is clear that for

we have

Now note that

and therefore for

We also have

and thus

Combining this with (B.5), we get

Now using the facts that for

and for

we arrive at

and hence

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

and

for any

Now for any

and

Therefore (combining with (B.5)),

and

Hence for any

for some

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

Therefore, in order to obtain the

Combining (B.2), (B.4), (B.13) and the fact that

This gives the desired

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

## Lemma B.2.

*There exist constants
*

*then there exists a solution ϕ of*

*such that*

*Here
*

## Proof.

Note that by the ε-regularity Theorem A.4 (with

for a.e.

## Lemma B.3.

*There exists a constant
*

*and u is a weakly intrinsic or extrinsic bi-harmonic map, then there exists
*

## 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

## Proof.

Let ϕ be a solution of (B.14), then

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