Conformally Invariant Variational Problems. - SAM

More documents

Recommendations

Info

Combining this to (VII.2) reveals that u ∞ is a solution of the CMC equation (VII.1). Obtaining information on the regularity of weak W 1,2 solutionsof the CMC equation(VII.2) requiressome more elaborate work. More precisely, a result from the theory of integration by compensation due to H. Wente is needed. Theorem VII.1. [We]Let a andbbe twofunctionsin W 1,2 (D 2 ), and let φ be the unique solution in W 1,p 0 (D 2 ) - for 1 ≤ p < 2 - of the equation ⎧ ⎨ −∆φ = ∂ x a∂ y b−∂ x b∂ y a in D 2 (VII.3) ⎩ ϕ = 0 on ∂D 2 . Then φ belongs to C 0 ∩W 1,2 (D 2 ) and ‖φ‖ L∞ (D 2 )+‖∇φ‖ L2 (D 2 ) ≤ C 0 ‖∇a‖ L2 (D 2 ) ‖∇b‖ L2 (D 2 ) . (VII.4) where C 0 is a constant independent of a and b. 12 ✷ Proof of theorem VII.1. Weshallfirstassumethataandb are smooth, so as to legitimize the various manipulations which we will need to perform. The conclusion of the theorem for general a and b in W 1,2 may then be reached through a simple density argument. In this fashion, we will obtain the continuity of φ from its being the uniform limit of smooth functions. Observe first that integration by parts and a simple application of the Cauchy-Schwarz inequality yields the estimate ∫ ∫ |∇φ| 2 = − φ∆φ ≤ ‖φ‖ ∞ ‖∂ x a∂ y b−∂ x b∂ y a‖ 1 D 2 D 2 ≤ 2‖φ‖ ∞ ‖∇a‖ 2 ‖∇b‖ 2 . 12 Actually, one shows that theorem VII.1 may be generalized to arbitrary oriented Riemannian surfaces, with a constant C 0 independent of the surface, which is quite a remarkable and useful fact. For more details, see [Ge] and [To]. 58
Accordingly, if φ lies in L ∞ , then it automatically lies in W 1,2 . Step 1. given two functions ã and ˜b in C0 ∞ (C), which is dense in W 1,2 (C), we first establish the estimate (VII.4) for ˜φ := 1 2π log 1 [ ] r ∗ ∂ x ã∂ y˜b−∂x˜b∂y ã . (VII.5) Owing to the translation-invariance, it suffices to show that We have ˜φ(0) = − 1 2π = − 1 2π = 1 2π |˜φ(0)| ≤ C 0 ‖∇ã‖ L2 (C) ‖∇˜b‖ L2 (C) . ∫ ∫ 2π 0 0 ∫ 2π ∫ +∞ 0 logr ∂ x ã∂ y˜b−∂x˜b∂y ã R 2 ∫ ( ) ( ) +∞ log r ∂ ã ∂˜b − ∂ ã ∂˜b ∂r ∂θ ∂θ ∂r 0 ã ∂˜b ∂θ dr r dθ (VII.6) dr dθ Because ∫ 2π ∂˜b 0 ∂θ dθ = 0, we may deduct from each circle ∂B r(0) a constant à ã chosen to have average ã r on ∂B r (0). Hence, there holds ˜φ(0) = 1 ∫ 2π ∫ +∞ [ã−ã r ] ∂˜b dr 2π 0 0 ∂θ r dθ . ApplyingsuccessivelytheCauchy-SchwarzandPoincaréinequalities on the circle S 1 , we obtain ⎛ |˜φ(0)| ≤ 1 ∫ +∞ (∫ dr 2π ) 1 ∫ |ã−ã r | 2 2 2π ⎝ ∂˜b ⎞ 1 2 2 ⎠ 2π 0 r 0 0 ∣∂θ∣ ≤ 1 ∫ ( ⎛ +∞ ∫ dr 2π ∂ã 2 )1 2 ∫ 2π ⎝ ∂˜b ⎞ 1 2 2 2π 0 r ∣ 0 ∂θ∣ ⎠ 0 ∣∂θ∣ The sought after inequality (VII.6) may then be inferred from the latter via applying once more the Cauchy-Schwarz inequality. 59
Page 1 and 2:
Conformally Invariant Variational P
Page 3 and 4:
II Conformal transformations - some
Page 5 and 6:
this implies ∀X,Y ∈ R m \{0} if
Page 7 and 8: This form is called the Hopf differ
Page 9 and 10: ∂ ∂x k (e 2λ δ ij ) = ∂ ∂
Page 11 and 12: Let k ∈ {1,...,n} and choose i
Page 13 and 14: If neither A = 0 nor B = 0 The left
Page 15 and 16: ✷ Proof of Lemma II.4 Since u(0)
Page 17 and 18: III Elementary Differential geometr
Page 19 and 20: control it gives on the image itsel
Page 21 and 22: This is the main advantage of worki
Page 23 and 24: oundary, and sending ∂D 2 monoton
Page 25 and 26: Such a Jordan curve is also simply
Page 27 and 28: 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: VII Integrabilitybycompensationtheo
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 and 80: Theorem VIII.2. [Riv1] Let N n be a
Page 81 and 82: If A is almost everywhere invertibl
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
Page 131 and 132:
dle to Φ(Σ ⃗ 2 ) : for all X
Page 133 and 134:
Let TR m ⃗ Φ([0,1]×Σ 2 ) be th
Page 135 and 136:
imply D ∂ ⃗H = 1 ∂t 2 2∑ D
Page 137 and 138:
We have moreover ∇ es ⃗ V N = =
Page 139 and 140:
for any perturbation V ⃗ which is
Page 141 and 142:
the Willmore Lagrangian ”controls
Page 143 and 144:
written in isothermal coordinates.
Page 145 and 146:
quantity div [ 2∇H ⃗ −3H∇
Page 147 and 148:
where e λ = |∂ x1Φ| ⃗ = |∂x
Page 149 and 150:
Observe that ⎧⎪ ⎨ ⎪ ⎩ 〈
Page 151 and 152:
and consequently ⋆(⃗n∧∇ ⊥
Page 153 and 154:
In one hand the projection into the
Page 155 and 156:
The tangential projection gives 4e
Page 157 and 158:
X.6 Construction of Isothermal Coor
Page 159 and 160:
the Lie algebra iR. Sections of thi
Page 161 and 162:
R m is given by ∇ X σ := π T (d
Page 163 and 164:
Let Σ 2 be a smoothcompactoriented
Page 165 and 166:
There exists a constant C > 0, such
Page 167 and 168:
such a way that ⃗n ρ λ realizes
Page 169 and 170:
We make now use of (X.87) and we de
Page 171 and 172:
We are now in positionto start the
Page 173 and 174:
isabilipschitzdiffeomorphismbetween
Page 175 and 176:
Having defined weak Willmore immers
Page 177 and 178:
Let µ be the solution of ⎧ ⎨
Page 179 and 180:
Combining this inequality with (X.1
Page 181 and 182:
ii) iii) iv) limsupArea( Φ ⃗ k (
Page 183 and 184:
The assumption i) implies that, mod
Page 185 and 186:
Theorem X.14. Let ⃗ Φ be a Lipsc
Page 187 and 188:
Thus < ∇ ⃗ Φ,∇ ⊥ ⃗ L >=
Page 189 and 190:
To that purpose we first compute
Page 191 and 192:
and the following equation holds
Page 193 and 194:
One verifies easily that π T ( ⃗
Page 195 and 196:
From (X.197) we compute ⋆(⃗n
Page 197 and 198:
For any such a vector field X satis
Page 199 and 200:
X.7.4 The conformal Willmore surfac
Page 201 and 202:
Denote ∂ z ⃗ L = A⃗ez +B⃗e
Page 203 and 204:
We have using (X.113) ∂ z ∂ z
Page 205 and 206:
References [Ad] Adams, David R. ”
Page 207 and 208:
[DHKW1] Dierkes, Ulrich; Hildebrand
Page 209 and 210:
larityof weak solutionsof nonlinear
Page 211 and 212:
[Poi] Poisson, Siméon Denis ”Ext
show all

Conformally Invariant Variational Problems. - SAM

Create successful ePaper yourself

Delete template?

Save as template?