Partial regularity for harmonic maps and related problems

Rivière, Tristan; Struwe, Michaël

doi:10.1002/cpa.20205

Cited by 98 publications

(167 citation statements)

References 13 publications

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“…The interior regularity was proved by T. Rivière in [Riv07] (for e ≡ 0) and our proof is based on Rivière's decomposition result combined with the Dirichlet growth approach by Rivière and Struwe in [RS08] as well as some additional arguments due to P. Strzelecki [Str03]. Remark 1.3 Let us emphasize that one can prove Theorem 1.1 also by reflection across ∂D 2 , whenever there is some ψ ∈ W 2,p (D 2 , R m ), p > 1, such that u = ψ on ∂D 2 .…”

Section: Remark 12mentioning

confidence: 86%

Boundary regularity via Uhlenbeck–Rivière decomposition

Müller¹,

Schikorra²

2009

Analysis

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We prove that weak solutions of systems with skew-symmetric structure, which possess a continuous boundary trace, have to be continuous up to the boundary. This applies, e.g., to the H-surface system u = 2H(u)∂ x 1 u ∧ ∂ x 2 u with bounded H and thus extends an earlier result by P. Strzelecki and proves the natural counterpart of a conjecture by E. Heinz. Methodically, we use estimates below natural exponents of integrability and a recent decomposition result by T. Rivière.

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Section: Remark 12mentioning

confidence: 86%

Boundary regularity via Uhlenbeck–Rivière decomposition

Müller¹,

Schikorra²

2009

Analysis

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“…By Theorem 1.2 in [30] which is developed from the regularity theory for critical elliptic systems with an anti-symmetric structure in [25,26,28,29,32], we know that v ∈ W 2, p (D + ρ (0)) for some ρ > 0 and any 1 < p < 2. In fact, the anti-symmetric term in the equation for v equals to zero.…”

Section: Theorem 28 Supposementioning

confidence: 99%

“…Integrating from 2 to L, we arrive at 26) where the last inequality follows from the energy identity (3.21). By using Lemma 2.1, now it is easy to deduce (3.24) from (3.22) and the above estimates (3.26) for energy decay.…”

Section: ) Up To a Subsequence Which Is Still Denoted By {(U N V mentioning

confidence: 99%

Bubbling analysis for approximate Lorentzian harmonic maps from Riemann surfaces

et al. 2017

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For a sequence of approximate harmonic maps (u n , v n ) (meaning that they satisfy the harmonic system up to controlled error terms) from a compact Riemann surface with smooth boundary to a standard static Lorentzian manifold with bounded energy, we prove that identities for the Lorentzian energy hold during the blow-up process. In particular, in the special case where the Lorentzian target metric is of the form g N −βdt 2 for some Riemannian metric g N and some positive function β on N , we prove that such identities also hold for the positive energy (obtained by changing the sign of the negative part of the Lorentzian energy) and there is no neck between the limit map and the bubbles. As an application, we complete the blow-up picture of singularities for a harmonic map flow into a standard static Lorentzian manifold. We prove that the energy identities of the flow hold at both finite and infinite singular times. Moreover, the no neck property of the flow at infinite singular time is true.

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“…Here, a particular skew symmetry of the nonlinear term in the Euler-Lagrange Communicated by L. Ambrosio. equations could be systematically exploited and generalized in the work of Hélein, Rivière and Struwe, see [14,[21][22][23]. This is also our starting point, both conceptually-because we generalize the harmonic map problem-and methodologically-because we shall use their techniques.…”

Section: Introductionmentioning

confidence: 99%

“…Here we explore this issue. We shall combine the regularity theory of [21][22][23] with Morrey space theory and a subtle iteration argument to achieve what should be the optimal regularity results in our setting.…”

Section: Introductionmentioning

confidence: 99%