Conservation laws for even order elliptic systems in the critical dimension -- a new approach

We consider elliptic systems in dimension $2m$. De Longueville and Gastel studied conservation laws for weak solutions of these systems generalizing the work of Rivi\`ere and Lamm-Rivi\`ere on second and fourth order systems. Using this they showed continuity of weak solutions. We follow a different approach by using a small perturbation of Uhlenbeck's gauge fixing matrix to construct a conservation law.


Introduction
In [7] Rivière discovered a method to write a two-dimensional conformally invariant non-linear elliptic PDE in divergence form. Then he used this conservation law to prove continuity of weak solutions. A well known example of a PDE satisfying his assumptions is the harmonic map equation. In [8] Rivière and Struwe explored this example further and gave a new proof of partial regularity for harmonic maps in higher dimensions. Later, the second author and Rivière [6] applied the same procedure to fourth order elliptic systems in four dimensions which are modeled on the biharmonic map equation (see also [10]). De Longueville and Gastel [2] recently extended this result to systems of order 2m in critical dimension. Theorem 1.1 (Thm 1.1-1.5 in [6] and Thm 4.1 in [2]). Assume m ≥ 2, n ∈ N. Let coefficient functions be given as for k ∈ {0, ..., m − 2}, V k ∈ W 2k+1−m,2 (B 2m , R n×n ⊗ ∧ 1 R 2m ) for k ∈ {0, ..., m − 1}, where V 0 = dη + F, η ∈ W 2−m,2 (B 2m , so(n)), F ∈ W 2−m, 2m m+1 ,1 (B 2m , R n×n ⊗ ∧ 1 R 2m ) We consider the equation ∆ k δ(w k du).  .
Our idea is to replace A by a small perturbation of the skew-symmetric matrix P in Uhlenbeck's gauge theorem. Theorem 1.2 (Thm 2.4 in [2]). Assume that m, n ∈ N and B r ⊂ R 2m is a ball of radius r. Then there is ε > 0 such that for all Ω ∈ W m−1,2 (B r , so(n) ⊗ ∧ 1 R 2m ) satisfying there are functions P ∈ W m,2 (B r/2 ; SO(n)) and ξ ∈ W m,2 (B r/2 , so(n) ⊗ ∧ 2 R 2m ) such that holds on B r/2 . Moreover, we have the estimate Rivière used this Ansatz in ( [9], p. 93-94) to derive a conservation law for solutions of ∆u = Ω · du, (1.4) where Ω ∈ L 2 (B 2 , so(n) ⊗ ∧ 1 R 2 ). Note that the Hodge-Laplace operator for ω : and for a 0-form f and a 1-form h for B ∈ W 1,2 (B 2 ) with δB = dεP + (id + ε)(dP + P Ω). We use the same idea and derive a conservation law for solutions of (1.1) with m ≥ 2.
Let coefficient functions be given as We consider the equation For this equation, the following statements hold.

(1.6)
There is σ 0 > 0 such that whenever σ < σ 0 , there exists a function ε ∈ W m,2 ∩ L ∞ (B 2m 1/2 ; M (n)) with ||ε|| W m,2 (B 2m 1/2 ) + ||ε|| L ∞ (B 2m 1/2 ) ≤ cσ and a distribution B ∈ W 2−m,2 (ii) A function u ∈ W m,2 (B 2m 1/2 , R n ) solves (1.5) weakly if and only if it is a distributional solution of the conservation law We start with the system (1.5) and decompose V 0 into V 0 = dη + F as in [2]. Following the work of de Longueville and Gastel in the proof of Theorem 4.1 (i) in [2] we find Ω ∈ W m−1,2 (B 2m , so(m) ⊗ ∧ 1 R 2m ) by repeatedly solving Neumann problems such that For σ > 0 sufficiently small we apply Theorem 1.2, a suitable version of Uhlenbeck's gauge theorem, and get ξ ∈ W m, 1/2 , M (n)). We multiply (1.5) with (id + ε)P and calculate We take a closer look at d∆ m−1 ((id + ε)P ). First note that we can rewrite the highest order term (id + ε)d∆ m−1 P so that it cancels (id + ε)P V 0 in (2.4). To see this we use (2.2), (2.1) as well as for a p-form ω and * * : (see [5]) Then we have where the c j ∈ N, 1 ≤ j ≤ 2m − 2 are constants and Now that we have removed the "worst" terms including V 0 , we want to examine this equation further and take a closer look at the function spaces of the summands. We separate the ε component from the rest and use the embedding results in Lemma B.1 and Lemma B.2 repeatedly. For the first term we have where we used the notation D k A ⋆ D l B for any linear combination of D k A and D l B, and D denotes the full derivative. The second term lies in The third and fourth terms are of the from where we used Lemma B.2 in the first step and Lemma B.
The fifth term follows in the same way For the last two terms we apply again Lemma B.2 and Lemma B.
Observe that all terms on the right-hand side of (2.5) consist of products W m−j,2 · W j+1−m,2 , j = 1, ..., 2m − 2 and L ∞ · W 2−m, 2m m+1 ,1 . Thus we can simplify (2.5) further and write However the equation still contains distributions. To take care of these we apply the same technique as de Longueville and Gastel and use the representation of negative Sobolev-Lorentz spaces.
This representation even yields a solution ε ∈ W m+1, 2m m+1 ,1 and this space embeds into W m,2 ∩ L ∞ . Together with (2.7) we get This representation allows us to shift derivatives away from the distributional part. For j = 1, ..., m − 2 we get The case j = 0 follows analogously We rewrite the left-hand side of (2.6) in the same way.
For the last term note that P ∈ W m,2 (B 2m 1/2 , SO(n)). Thus we identify P with K 2m−1 and write Putting all of this together we get an equation equivalent to (2.4) We simplify this further by setting for every γ with |γ| ≤ m − 2. To see this last inequality we use (2.8) and estimate each term separately.
With this we have Choosing σ < min{ µ 2c1(µ+1) , 1 2c2 } shows that Ψ is a contraction. Now we can apply the Banach fixed point theorem which yields a uniqueε * ∈ X solving (2.11) and by Lemma A.1 and (2.10) Thus we have for every γ with |γ| ≤ m − 2. We want to reverse the abbreviations we made at the beginning to get a detailed look at (2.13). To do this we go back to (2.5). As we have seen before, each term of this equation is a product of a distribution and a Sobolev function. More precisely, the terms are of the form L ∞ · W 2−m, 2m m−1 and W m−k,2 · W −m+1+k,2 , k = 1, ..., 2m − 2. We use the following representations for the distributions Then we shift derivatives to get an equation of the form |γ|≤m−2 ∂ γ (...) γ = 0 as in (2.9). Using this we see that (2.13) is equivalent to By Lemma 10.68 in [3] there exist B γ ∈ W because (2.5) and (2.4) are equivalent and This finishes the proof.
Appendix A.