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Creators/Authors contains: "Mazowiecka, Katarzyna"

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  1. ; ; (, Memoirs of the American Mathematical Society)
    We consider minimizing harmonic maps u u from Ω<#comment/> ⊂<#comment/> R n \Omega \subset \mathbb {R}^n into a closed Riemannian manifold N \mathcal {N} and prove: 1. an extension to n ≥<#comment/> 4 n \geq 4 of Almgren and Lieb’s linear law. That is, if the fundamental group of the target manifold N \mathcal {N} is finite, we have\[ H n −<#comment/> 3 ( sing ⁡<#comment/> u ) ≤<#comment/> C ∫<#comment/> ∂<#comment/> Ω<#comment/> | ∇<#comment/> T u | n −<#comment/> 1 d H n −<#comment/> 1 ; \mathcal {H}^{n-3}(\operatorname {sing} u) \le C \int _{\partial \Omega } |\nabla _T u|^{n-1} \,\mathrm {d}\mathcal {H}^{n-1}; \]2. an extension of Hardt and Lin’s stability theorem. Namely, assuming that the target manifold is N = S 2 \mathcal {N}=\mathbb {S}^2 we obtain that the singular set of u u is stable under small W 1 , n −<#comment/> 1 W^{1,n-1} -perturbations of the boundary data. In dimension n = 3 n=3 both results are shown to hold with weaker hypotheses, i.e., only assuming that the trace of our map lies in the fractional space W s , p W^{s,p} with s ∈<#comment/> ( 1 2 , 1 ] s \in (\frac {1}{2},1] and p ∈<#comment/> [ 2 , ∞<#comment/> ) p \in [2,\infty ) satisfying s p ≥<#comment/> 2 sp \geq 2 . We also discuss sharpness. 
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  2. ; (, Journal of the London Mathematical Society)
    Abstract Let a closed ‐dimensional manifold, be a closed manifold, and let for . We extend the monumental work of Sacks and Uhlenbeck by proving that if , then there exists a minimizing ‐harmonic map homotopic to . If , then we prove that there exists a ‐harmonic map from to in a generating set of . Since several techniques, especially Pohozaev‐type arguments, are unknown in the fractional framework (in particular, when , one cannot argue via an extension method), we develop crucial new tools that are interesting on their own: such as a removability result for point singularities and a balanced energy estimate for nonscaling invariant energies. Moreover, we prove the regularity theory for minimizing ‐maps into manifolds. 
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