Conformally Invariant Variational Problems. - SAM

More documents

Recommendations

Info

VI <strong>Conformally</strong>invariantcoerciceLagrangians with quadratic growth, in dimension 2. The resolution of the Plateau problem proposed by Douglas and Radó is an example of the use of a conformal invariant Lagrangian E to approach an “extrinsic” problem: minimizing the area of a disk with fixed boundary. The analysis of this problem was eased by the high simplicity of the equation (V.4) satisfied by the critical points of E. It is the Laplace equation. Hence, questions related to unicity, regularity, compactness, etc... can be handled with a direct application of the maximum principle. In the coming three chapters, we will be concerned with analogous problems (in particular regularity issues) related to the critical points to general conformally invariant, coercive Lagrangians with quadratic growth. As we will discover, the maximum principle no longer holds, and one must seek an alternate way to compensate this lack. The conformal invariance of the Lagrangian will generate a very peculiar type of nonlinearities in the correspondingEuler-Lagrangeequations. We willsee how the specific structure of these nonlinearitiesenable one to recast the equations in divergence form. This new formulation, combined to the results of integration by compensation, will provide the substrate to understanding a variety of problems, such as Willmore surfaces, poly-harmonic and α-harmonic maps, Yang- Mills fields, Hermitte-Einstein equations, wave maps, etc... We consider a Lagrangian of the form ∫ L(u) = l(u,∇u) dx dy , (VI.1) D 2 where the integrand l is a function of the variables z ∈ R m and p ∈ R 2 ⊗R m , which satisfy the following coercivity and “almost quadratic” conditions in p: C −1 |p| 2 ≤ l(z,p) ≤ C |p| 2 , 42 (VI.2)
We further assume that L is conformally invariant: for each positive conformal transformation f of degree 1, and for each map u ∈ W 1,2 (D 2 ,R m ), there holds ∫ L(u◦f) = l(u◦f,∇(u◦f)) dx ′ dy ′ f −1 (D 2 ) ∫ (VI.3) = l(u,∇u) dx dy = L(u) . D 2 Example 1. The Dirichlet energy described in the Introduction, ∫ E(u) = |∇u| 2 dxdy , D 2 whose critical points satisfy the Laplace equation (V.4), which, owing to the conformalhypothesis, geometricallydescribes minimal surfaces. Regularity and compactness matters relative to this equation are handled with the help of the maximum principle. Example 2. LetanarbitraryinR m begiven,namely(g ij ) i,j∈Nm ∈ C 1 (R m ,S + m), whereS + m denotesthe subset of M m (R), comprising the symmetric positive definite m×m matrices. We make the following uniform coercivity and boundedness hypothesis: ∃ C > 0 s. t. C −1 δ ij ≤ g ij ≤ Cδ ij on R m . Finally, we suppose that ‖∇g‖ L∞ (R m ) < +∞ . With these conditions, the second example of quadratic, coercive, conformally invariant Lagrangian is E g (u) = 1 ∫ 〈∇u,∇u〉 2 g dxdy D 2 = 1 ∫ ∑ m g ij (u)∇u i·∇u j dxdy . 2 D 2 i,j=1 43
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: V.3 Existence of Parametric disc ex
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 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?