Harmonic Maps, Conservation Laws, And Moving Frames

By Frédéric Hélein. 2002

468 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/77f786085af99a916185cc97a29fa45c0ef0f7ce

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Sobolev Mappings between Manifolds and Metric Spaces (2009, background): “Here, A is a second fundamental form of the embedding of N into the Euclidean space (see, for example, [4, 16, 34, 41, 42, 48, 65, 69, 79, 80, 83, 84]).”
ANISOTROPIC NONLINEAR ELLIPTIC SYSTEMS WITH MEASURE DATA AND ANISOTROPIC HARMONIC MAPS INTO SPHERES (2006, methodology): “Questions like this have been studied by Chen [13], Shatah [38], Evans [21, 20], and Hélein [25] for harmonic maps, which are special cases (p = 2) of p-harmonic maps (see also earlier work by Schoen and Uhlenbeck [36] on minimizing maps).”
Rough solutions for the wave maps equation (2005, background): “This idea comes from previous work on harmonic maps (see Hélein’s book [7] and references therein) but its implementation for the wave maps equation is much more delicate.”
The wave maps equation (2004, background): “The harmonic maps have their own story (see [8] and references therein) and at some point also had an impact on the study of wave maps.”
Global Existence of Weak Solutions of the Nematic Liquid Crystal Flow in Dimension Three (2014, unknown): “ |∇Hǫ|2 = − ˆ R3 dde ǫ ×ddeǫ · Hǫ = − ˆ R3 dde ǫ×deǫ · d∗Hǫ = − ˆ R3 dωeǫ × (fe2eωǫ) ·d∗Hǫ. 8 F. LIN AND C. WANG Applying the duality between the Hardy space H1(R3) and the BMO space BMO(R3) (see [4] [6] or [20]), we then obtain ˆ R3 |∇Hǫ|2 . deω ǫ ·d ∗Hǫ H1(R3) fe2 ǫ ωeǫ BMO(R3) . ∇eω ǫ L2(R3) ∇Hǫ ”
An Approximation Scheme for Schrödinger Maps (2007, background): “(See the discussion of the space H1 in the book by Hélein, 2002).”
Harmonic maps from degenerating Riemann surfaces (2008, background): “3) weakly is smooth ([10], [11], or [28] for a new proof).”
Existence and regularity of maximal metrics for the first Laplace eigenvalue on surfaces (2013, background): “Then Φ is weakly harmonic and by the regularity theory of harmonic functions by Hélein (see [19]), we can complete the proof of the theorem.”
Partial regularity for biharmonic maps, revisited (2008, unknown): “(7) Observing that w j∇u j = 0, following Hélein [5] we rewrite ((w ⊗ w) u)i = wiw j u j = −wi∇w j · ∇u j = (w j∇wi − wi∇w j ) · ∇u j , (8) where we tacitly sum over repeated indices.”
Decay estimates for Rivière’s equation, with applications to regularity and compactness (2011, background): “In particular for f ∈ H(1)(R(2)), writing w = N [f ], we have the estimate (see [9])”