Abstract
Following Rivière’s study of conservation laws for second order quasilinear systems with critical nonlinearity and Lamm/Rivière’s generalization to fourth order, we consider similar systems of order 2m. Typical examples are m-polyharmonic maps. Under natural conditions, we find a conservation law for weak solutions on 2m-dimensional domains. This implies continuity of weak solutions.
Highlights
The regularity of harmonic maps and related systems has been an active topic of research for decades
The bit of extra structure for Ω from the harmonic map system is that it can be rewritten in such a way that Ω takes its values in the skew-symmetric matrices
Note that the regularity assumption on the least regular coefficient V0 is quite analogous to those made above. It is slightly better than the V0 ∈ W 2−m,2 one would have in the general critical nonlinearity, in the sense that it decomposes in a skew-symmetric term and a term enjoying a tiny bit of extra (Lorentz) regularity
Summary
The regularity of harmonic maps and related systems has been an active topic of research for decades. The bit of extra structure for Ω from the harmonic map system is that it can be rewritten in such a way that Ω takes its values in the skew-symmetric matrices This is what is needed to prove continuity of weak solutions. Note that the regularity assumption on the least regular coefficient V0 is quite analogous to those made above It is slightly better than the V0 ∈ W 2−m,2 one would have in the general critical nonlinearity, in the sense that it decomposes in a skew-symmetric term and a term enjoying a tiny bit of extra (Lorentz) regularity. Η and several other coefficients are in Sobolev spaces of negative order, which means they only make sense as distributions This results in a conservation law that depends on a function A ∈ W m,2 ∩ L∞ and a distribution B ∈ W 2−m,2 that satisfy some auxiliary equation. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) — 262992434
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