ϵ-regularity for systems involving non-local, antisymmetric operators
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Submission date: 15. Jul. 2012
published in: Calculus of variations and partial differential equations, 54 (2015) 4, p. 3531-3570
DOI number (of the published article): 10.1007/s00526-015-0913-3
MSC-Numbers: 58E20, 35B65, 35J60
Keywords and phrases: harmonic maps, nonlinear elliptic pde, regularity of solutions
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We prove an epsilon-regularity theorem for critical and super-critical systems with a non-local antisymmetric operator on the right-hand side.
These systems contain as special cases, both, Euler-Lagrange equations of conformally invariant variational functionals as Riviere treated them, and also Euler-Lagrange equations of fractional harmonic maps introduced by Da Lio-Riviere.
In particular, the arguments give new and uniform proofs of the regularity results by Riviere, Riviere-Struwe, Da-Lio-Riviere, and also the integrability results by Sharp-Topping and Sharp, not discriminating between the classical local, and the non-local situations.
One important ingredient for this kind of stability, relies on the proof of uniformity of Hardy-space estimates for bi-commutators as the leading differential order goes to two.