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A compactness result for polyharmonic maps in the critical dimension

Shenzhou Zheng (2016)

Czechoslovak Mathematical Journal

For n = 2 m 4 , let Ω n be a bounded smooth domain and 𝒩 L a compact smooth Riemannian manifold without boundary. Suppose that { u k } W m , 2 ( Ω , 𝒩 ) is a sequence of weak solutions in the critical dimension to the perturbed m -polyharmonic maps d d t | t = 0 E m ( Π ( u + t ξ ) ) = 0 with Φ k 0 in ( W m , 2 ( Ω , 𝒩 ) ) * and u k u weakly in W m , 2 ( Ω , 𝒩 ) . Then u is an m -polyharmonic map. In particular, the space of m -polyharmonic maps is sequentially compact for the weak- W m , 2 topology.

New estimates for elliptic equations and Hodge type systems

Jean Bourgain, Haïm Brezis (2007)

Journal of the European Mathematical Society

We establish new estimates for the Laplacian, the div-curl system, and more general Hodge systems in arbitrary dimension n , with data in L 1 . We also present related results concerning differential forms with coefficients in the limiting Sobolev space W 1 , n .

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