Conformally Invariant Variational Problems. - SAM

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holds m∑ A j ik ∇uj = 0 ∀ i,k = 1···m . (VIII.9) j=1 Inserting this identity into (VIII.8) produces m∑ ∆u i = − (Hjk(u)−H i j ik (u)) ∇⊥ u k ·∇u j − j,k=1 m∑ (A i jk (u)−Aj ik (u)) ∇uk ·∇u j . j,k=1 (VIII.10) The m×m matrix Ω := (Ω i j ) i,j=1···m defined via m Ω i j := ∑ (Hjk i (u)−Hj ik (u))∇⊥ u k + k=1 m∑ (A i jk (u)−Aj ik (u))∇uk , isevidentlyantisymmetric,anditbelongstoL 2 . Withthisnotation, (VIII.10) is recast in the form (VIII.6), and thus all of the hypotheses of theorem VIII.1 are fulfilled, thereby concluding the proof of theorem VIII.2. ✷ On the conservation laws for Schrödinger systems with antisymmetric potentials. Per the above discussion, there only remains to establish theorem VIII.1 in order to reach our goal. To this end, we will express the Schrödinger systems with antisymmetric potentials in the form of conservation laws. More precisely, we have Theorem VIII.3. [Riv1] Let Ω be a matrix-valued vector field on D 2 in L 2 (∧ 1 D 2 ,so(m)). Suppose that A and B are two W 1,2 functions on D 2 taking their values in the same of square m×m matrices which satisfy the equation k=1 ∇A−AΩ = −∇ ⊥ B . (VIII.11) 80
If A is almost everywhere invertible, and if it has the bound ‖A‖ L∞ (D 2 ) +‖A −1 ‖ L∞ (D 2 ) < +∞ , (VIII.12) then u is a solution of the Schrödinger system (VIII.6) if and only if it satisfies the conservation law div(A∇u−B∇ ⊥ u) = 0 . (VIII.13) If (VIII.13) holds, then u ∈ W 1,p loc (D2 ,R m ) for any 1 ≤ p < +∞, and therefore u is Hölder continuous in the interior of D 2 , C 0,α loc (D2 ) for any α < +∞ . ✷ We note that the conservation law (VIII.13), when it exists, generalizes the conservation laws previously encountered in the study of problems with symmetry, namely: 1) In the case of the constant mean curvature equation, the conservation law (VII.1) is (VIII.13) with the choice A ij = δ ij , and ⎛ B = ⎜ ⎝ 0 −Hu 3 Hu 2 Hu 3 0 −Hu 1 ⎞ ⎟ ⎠ −Hu 2 Hu 1 0 2) In the case of S n -valued harmonic maps, the conservation law (VII.25) is (VIII.13) for A ij = δ ij , and B = (B i j ) with ∇ ⊥ B i j = u i ∇u j −u j ∇u i . Theultimatepartofthissectionwillbedevotedtoconstructing A and B, for any given antisymmetric Ω, with sufficiently 81
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Conformally Invariant Variational P
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II Conformal transformations - some
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this implies ∀X,Y ∈ R m \{0} if
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This form is called the Hopf differ
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∂ ∂x k (e 2λ δ ij ) = ∂ ∂
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Let k ∈ {1,...,n} and choose i
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If neither A = 0 nor B = 0 The left
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✷ Proof of Lemma II.4 Since u(0)
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III Elementary Differential geometr
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control it gives on the image itsel
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This is the main advantage of worki
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oundary, and sending ∂D 2 monoton
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Such a Jordan curve is also simply
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Therefore, for m = 3, any minimizer
Page 29 and 30: critical point for variations in th
Page 31 and 32: Remark V.3. There are situations wh
Page 33 and 34: Thus the flow x t of the vector-fie
Page 35 and 36: Another difficulty lies in the rema
Page 37 and 38: and ‖u(x)−u(y)‖ 2 L ∞ ((∂
Page 39 and 40: in the middle of this arc : |p −
Page 41 and 42: V.3 Existence of Parametric disc ex
Page 43 and 44: We further assume that L is conform
Page 45 and 46: We note that Γ i (∇u,∇u) :=
Page 47 and 48: is explicitly given by u(x,y) := lo
Page 49 and 50: Example 3. We consider a map (ω ij
Page 51 and 52: norm. The analytical difficulties r
Page 53 and 54: acting on maps u ∈ W 1,2 (D 2 ,N
Page 55 and 56: Whence, [ ∆u−H(u)(∇ ⊥ u,∇
Page 57 and 58: VII Integrabilitybycompensationtheo
Page 59 and 60: Accordingly, if φ lies in L ∞ ,
Page 61 and 62: Proof of the regularity of the solu
Page 63 and 64: If wemultiplytheLaplaceequationthro
Page 65 and 66: where u still denotes the normal un
Page 67 and 68: frames, thereby compensating for th
Page 69 and 70: tangent frame field to T 2 . Define
Page 71 and 72: egularity. Note that (VI.26) is equ
Page 73 and 74: Just as in the proof of the regular
Page 75 and 76: VIII A proof of Heinz-Hildebrandt
Page 77 and 78: maps, namely ∑n+1 ( −∆u i =
Page 79: Theorem VIII.2. [Riv1] Let N n be a
Page 83 and 84: the whole regularity result stated
Page 85 and 86: Bringing altogether (VIII.15), (VII
Page 87 and 88: In the simpler case when Ω is dive
Page 89 and 90: Theorem VIII.5. [Uhl], [Riv1] Let m
Page 91 and 92: yields the existence of the solutio
Page 93 and 94: IX A PDE version of the constant va
Page 95 and 96: well known variation of the constan
Page 97 and 98: elliptic estimates give ‖∆ −1
Page 99 and 100: a meaning to (IX.61) we need at lea
Page 101 and 102: one, similar approaches can be very
Page 103 and 104: form associated to g on S at the po
Page 105 and 106: where g(S) denotes the genus of S.
Page 107 and 108: - General relativity : The Willmore
Page 109 and 110: where ⋆ is the Hodge operator 37
Page 111 and 112: Let us take locally about p a norma
Page 113 and 114: and that we have denoted by ( ⃗
Page 115 and 116: and similarly the second fundamenta
Page 117 and 118: Let (⃗n 1 ,··· ,⃗n m−2 ) b
Page 119 and 120: orthonormal basis for the metric g.
Page 121 and 122: X.4.2 Li-Yau Energy lower bounds an
Page 123 and 124: Thus ⃗G ∗ ω S m−1(p) = 1 |S
Page 125 and 126: the half space given by the affine
Page 127 and 128: where we used that the unit sphere
Page 129 and 130: Conjecture X.1. Let ⃗ Φ be an im
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dle to Φ(Σ ⃗ 2 ) : for all X
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Let TR m ⃗ Φ([0,1]×Σ 2 ) be th
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imply D ∂ ⃗H = 1 ∂t 2 2∑ D
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We have moreover ∇ es ⃗ V N = =
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for any perturbation V ⃗ which is
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the Willmore Lagrangian ”controls
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written in isothermal coordinates.
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quantity div [ 2∇H ⃗ −3H∇
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where e λ = |∂ x1Φ| ⃗ = |∂x
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Observe that ⎧⎪ ⎨ ⎪ ⎩ 〈
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and consequently ⋆(⃗n∧∇ ⊥
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In one hand the projection into the
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The tangential projection gives 4e
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X.6 Construction of Isothermal Coor
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the Lie algebra iR. Sections of thi
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R m is given by ∇ X σ := π T (d
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Let Σ 2 be a smoothcompactoriented
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There exists a constant C > 0, such
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such a way that ⃗n ρ λ realizes
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We make now use of (X.87) and we de
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We are now in positionto start the
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isabilipschitzdiffeomorphismbetween
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Having defined weak Willmore immers
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Let µ be the solution of ⎧ ⎨
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Combining this inequality with (X.1
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ii) iii) iv) limsupArea( Φ ⃗ k (
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The assumption i) implies that, mod
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Theorem X.14. Let ⃗ Φ be a Lipsc
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Thus < ∇ ⃗ Φ,∇ ⊥ ⃗ L >=
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To that purpose we first compute
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and the following equation holds
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One verifies easily that π T ( ⃗
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From (X.197) we compute ⋆(⃗n
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For any such a vector field X satis
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X.7.4 The conformal Willmore surfac
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Denote ∂ z ⃗ L = A⃗ez +B⃗e
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We have using (X.113) ∂ z ∂ z
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References [Ad] Adams, David R. ”
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[DHKW1] Dierkes, Ulrich; Hildebrand
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larityof weak solutionsof nonlinear
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[Poi] Poisson, Siméon Denis ”Ext
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