Travaux sur les symÃ©tries de Lie des Ã©quations aux ... - DMA - Ens

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368possible modified Jacobian determinants:(2.27)⎧⎪⎨⎪⎩∆(x 1 x 1 |x 2 |y) ∆(x 1 |x 1 x 1 |y) ∆(x 1 |x 2 |x 1 x 1 )∆(x 1 x 2 |x 2 |y) ∆(x 1 |x 1 x 2 |y) ∆(x 1 |x 2 |x 1 x 2 )∆(x 1 y|x 2 |y) ∆(x 1 |x 1 y|y) ∆(x 1 |x 2 |x 1 y)∆(x 2 x 2 |x 2 |y) ∆(x 1 |x 2 x 2 |y) ∆(x 1 |x 2 |x 2 x 2 )∆(x 2 y|x 2 |y) ∆(x 1 |x 2 y|y) ∆(x 1 |x 2 |x 2 y)∆(yy|x 2 |y) ∆(x 1 |yy|y) ∆(x 1 |x 2 |yy).Next, we observe that if we want to solve with respect to y x 1 x1 in (2.10),with respect to y x 1 x 2 in (2.22) and with respect to y x 2 x2 in (2.23), we have todivide by the Jacobian determinant ∆(x 1 |x 2 |y). Consequently, we introduce18 new square functions as follows:(2.28)⎧⎪⎨⎪⎩□ 1 x 1 x 1 := ∆(x1 x 1 |x 2 |y)∆(x 1 |x 2 |y)□ 1 x 2 x 2 := ∆(x2 x 2 |x 2 |y)∆(x 1 |x 2 |y)□ 2 x 1 x 1 := ∆(x1 |x 1 x 1 |y)∆(x 1 |x 2 |y)□ 2 x 2 x 2 := ∆(x1 |x 2 x 2 |y)∆(x 1 |x 2 |y)□ 3 x 1 x 1 := ∆(x1 |x 2 |x 1 x 1 )∆(x 1 |x 2 |y)□ 3 x 2 x 2 := ∆(x1 |x 2 |x 2 x 2 )∆(x 1 |x 2 |y)□ 1 x 1 x 2 := ∆(x1 x 2 |x 2 |y)∆(x 1 |x 2 |y)□ 1 x 2 y := ∆(x2 y|x 2 |y)∆(x 1 |x 2 |y)□ 2 x 1 x 2 := ∆(x1 |x 1 x 2 |y)∆(x 1 |x 2 |y)□ 2 x 2 y := ∆(x1 |x 2 y|y)∆(x 1 |x 2 |y)□ 3 x 1 x 2 := ∆(x1 |x 2 |x 1 x 2 )∆(x 1 |x 2 |y)□ 3 x 2 y := ∆(x1 |x 2 |x 2 y)∆(x 1 |x 2 |y)□ 1 x 1 y := ∆(x1 y|x 2 |y)∆(x 1 |x 2 |y)□ 1 yy := ∆(yy|x2 |y)∆(x 1 |x 2 |y)□ 2 x 1 y := ∆(x1 |x 1 y|y)∆(x 1 |x 2 |y)□ 2 yy := ∆(x1 |yy|y)∆(x 1 |x 2 |y)□ 3 x 1 y := ∆(x1 |x 2 |x 1 y)∆(x 1 |x 2 |y)□ 3 yy := ∆(x1 |x 2 |yy)∆(x 1 |x 2 |y) .Thanks to these notations, we can rewrite the three equations (2.10),(2.22) and (2.23) in a more compact style.Lemma 2.29. A completely integrable system of three second order partialdifferential equations(2.30)⎧⎪⎨y x 1 x 1(x) = F ( 1,1 x 1 , x 2 , y(x), y x 1(x), y x 2(x) ) ,y x⎪⎩1 x 2(x) = F ( 1,2 x 1 , x 2 , y(x), y x 1(x), y x 2(x) ) ,y x 2 x 2(x) = F ( 2,2 x 1 , x 2 , y(x), y x 1(x), y x 2(x) ) ,is equivalent to the simplest system Y X 1 X 1 = 0, Y X 1 X 2 = 0, Y X 2 ,X 2 = 0, ifand only if there exist local K-analytic functions X 1 , X 2 , Y such that it may
e written under the specific form:(2.31)⎧)y x 1 x 1 = −□3 x 1 x+ y 1 x 1 ·(−2□ 3 x 1 y + □1 x 1 x+ y 1 x 2 · (□ 2 )x 1 x +( ) ( )1+ y x 1 y x 1 · −□ 3 yy + 2□ 1 x 1 y+ y x 1 y x 2 · 2□ 2 x 1 y++ y x 1 y x 1 y x 1 · (□ 1 yy)+ yx 1 y x 1 y x 2 · (□ 2 yy),))y x⎪⎨1 x 2 = −□3 x 1 x+ y 2 x 1 ·(−□ 3 x 2 y + □1 x 1 x+ y 2 x 2 ·(−□ 3 x 1 y + □2 x 1 x+( ) ()2+ y x 1 y x 1 · □ 1 x 2 y+ y x 1 y x 2 · −□ 3 yy + □ 1 x 1 y + □2 x 2 y+( )+ y x 2 y x 2 · □ 2 x 1 y+ y x 1 y x 1 y x 2 · (□ 1 yy)+ yx 1 y x 2 y x 2 · (□ 2 yy),y x 2 x 2 = −□3 x 2 x+ y 2 x 1 · (□ 1 ) )x 2 x + 2 yx 2 ·(−2□ 3 x 2 y + □2 x 2 x+( ) (2 )+ y x 1 y x 2 · 2□ 1 x⎪⎩2 y+ y x 2 y x 2 · −□ 3 yy + 2□2 x 2 y++ y x 1 y x 2 y x 2 · (□ 1 yy)+ yx 2 y x 2 y x 2 · (□ 2 yy).3692.32. General formulas. The formal dictionary between the original determinantialformulas (2.10), (2.22), (2.23), between the coefficients (2.15) ofthe second order prolongation of a vector field and between the new squareformulas (2.31) above is evident. Consequently, without any computation,just by translating the family of formulas (2.14), we may deduce the exactformulation of the desired generalization of Lemma 2.29 above.Lemma 2.33. A completely integrable system of second order partial differentialequations of the form(2.34)y x j 1x j 2 (x) = F j 1,j 2(x, y(x), y x 1(x), . . .,y x n(x)) , j 1 , j 2 = 1, . . .n,is equivalent to the simplest system Y X j 1X j 2 = 0, j 1 , j 2 = 1, . . ., n, if andonly if there exist local K-analytic functions X l , Y such that it may be writtenunder the specific form:(2.35) ⎧⎪⎨⎪⎩y x j 1x j 2 = −□ n+1x j 1x j 2 + n∑+y x j 1 ·k 1 =1y x k 1 ·(□ k 1x j 2y − 1 )2 δk 1j 2□ n+1yy+y x j 1 y x j 2 · (□ k 1yy)}.{()□ k 1x j 1x − j 2 δk 1j 1□ n+1x j 2y − δk 1j 2□ n+1 +x j 1y+ y x j 2 ·(□ k 1x j 1y − 1 )2 δk 1j 1□ n+1yy +
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Joël M E R K E RÉcole Normale Sup
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§1. INTRODUCTIONSeveral physically
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e a collection of m analytic second
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7where the indices j, l 1 vary in {
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This phenomenon could be explained
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This yields the prolongation of the
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X(x, y) and Y (x, y) such that it m
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2.23. Compatibility conditions for
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This lemma is left to the reader; a
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simplification nor any reordering:(
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aleza, y por otra, las organizacion
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computational level (differential-g
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For instance, in the case m = 2, by
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in (3.11). The second equation that
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Similarly, the second equation take
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order derivatives of X and of the Y
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Lemma 3.32. The system yxx j = F j
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Lemma 3.45. The following quadratic
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often be denoted by the sign “·
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1Now, taking account of the factor
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conditions, totally equivalent to t
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43remaining terms afterwards:(3.71)
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Here, the sign ≡ precisely means:
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Next, replacing plainly (3.64) in (
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49the order of §3.73. We get:(3.89
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51Multiplying by −2 and reorganiz
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⎧Θ l 1y l 2 = −Ll 1l 1 ,l 1 ,y
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+ 1 ∑4 δj l 1Hl k 2L k k,kk− 2
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+ 1 ∑2 δj l 1Hl k 3M l2 ,kk+ 1 4
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In the hardest techical part of thi
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− 1 3+ 1 3− 1 4+ 1 4∑Hl k 1H
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− 1 3∑kL l 1l1 ,k,x Hk l 1+ 1 3
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Here, the sign ≡ means “modulo
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Thirdly, put j := l 2 in (3.108) wi
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69correct. We get:(4.29)0 =?== −
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Next, apply the operator ∑ k Ll 2
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73Writing term by term the substrac
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of the terms of the subgoal (4.29):
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77+ 1 2∑kH k l 1 ,y l 2 Hl 2k15
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79= − X xx Yx j + Y jm∑+++++l 1
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Our first task is to compute the de
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Replacing this expression of A k in
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85have the continuation(5.17) −y
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87and where thirdly (we are nearly
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89obtain:(5.23)III :=m∑m∑l 1 =1
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[GTW1989] GRISSOM, C.; THOMPSON, G.
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93Nonalgebraizable real analytic tu
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and of CR dimension m = n − d ≥
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coordinates z ′ = 2i ln(z/z p ),
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{x ∈ K n : |x| < ρ} for some ρ
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(2) A K-algebraic inversion mapping
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(1) The complex Lie algebra Hol(M,
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Proposition 3.1. Let t ↦→ G i (
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The first relation gives nothing, s
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with |β∗ k | ≥ 1 and integers
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Clearly, the left hand side is an a
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field X 1 ′ := h ∗ (X 1 ). Taki
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which yields after differentiating
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117Im w∆ n(ρ 4 )∆ n(ρ 1 )∆
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[∂ i,ei H ei (t ′ )] t ′ =He
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p = (w p , z p , ζ p , ξ p ) ∈
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Finite nondegeneracy is interesting
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such that |ω j i ∗ ,β∗ i(t,
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for k even, we have Γ k (z (k) ) =
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that for every local holomorphic se
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h(t) = ˜H(t, J l 0µ 0¯h(0)) with
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infinite families of pairwise non b
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functions w(z)). The coefficients R
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Then the Lie criterion states that
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y k ′ = λk k y k and w ′ = µ
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The last statements of Corollaries
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[Sha2000] SHAFIKOV, R.: Analytic co
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Here x = (x 1 , . . ., x n ) ∈ K
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Consequently, for the case κ = 2,
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are devoted to provide a general on
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for l = 1, . . .,m such that the lo
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Lemma 8.1. The following conditions
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155namely X ←→ X + X ←→ X .
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In terms of Sussmann’s approach [
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x κ−1 χ κ−1 + O(|x| κ ) + O
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Observe that these expressions are
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where the first term I involves onl
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I 5 = ∑ i 1I 6 = ∑ i 1 ,i 2I 7
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U 2 U κ−1 , we obtain the five f
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and U 2 U κ−1 , we obtain the fi
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following equations for j = 1, . .
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linear equations:(7.28) ⎧0 = Rx j
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175R j containing at least one part
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177[3] :∑σ∈S κ−1κδ l,....
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the values of the partial derivativ
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also appears some derivatives Q l
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[5] F. ENGEL; LIE, S.: Theorie der
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185Nonrigid sphericalreal analytic
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origin:{ ( ∣ ∣ )AJ 4 1∣∣∣
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I 2 characterizes equivalence to w
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y means of a fundamental, elementar
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2) When M is Levi nondegenerate at
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195holding in C { z ′ , z ′ , w
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Fortunately for our present purpose
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We notice passim that S ≡ Q x (no
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for a certain local K-analytic new
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Conversely, if y xx (x) = F ( x, y(
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205But we may also express the dual
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Then thanks to a straightforward ap
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and we then expand carefully the re
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explicit computation, so let us rew
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213+T ah3Q 2 a Q b∆(aa|b)∆(b|bb
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215Vanishing Hachtroudi curvaturean
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to fix the ideas, it will be assume
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then by replacing the(so obtained)v
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and also in addition the fact that
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and this defines without ambiguity
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the leaves of this local foliation
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Then by differentiating with respec
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∂Here, the coefficients of the2 T
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231for (7.28):( )∂□ ·∂a □
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233Lie symmetriesof partial differe
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Differentiating the first equation
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Example 1.21. (Continued) With n =
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defined by the graphed equations:((
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for some two local analytic maps Π
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Proof. Let l = l ( x i , y j , yβ(
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complementary views on the same obj
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Classification problem 3.11. Classi
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Definition 4.10. L is an infinitesi
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Convention 5.2. The letters R will
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Assuming that dimSYM(E 1 ) = 8, tak
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Restarting from §4.1, let ϕ a Lie
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Here, R j l 1 ,kare universal polyn
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Lemma 6.34. Let M be a submanifold
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where the expressions D β,β1 are
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Then the sum L + L is tangent to M
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Lemma 7.11. ([CM1974, BER1999, Me20
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where(7.33) ∆ := ∑1kn∂ 2∂x
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with r, s being unspecified functio
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and moreover, all other, higher ord
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To conclude, we replace X xx so obt
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Definition 8.43. A real analytic hy
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Lemma 8.59. The function b depends
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where the letter r denotes an unspe
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(8.80) ⎧Y 1,1,1 = Y x 1 x 1 x 1 +
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(X 1 , X 2 , Y , Xy 1,X 2x, Y 2 x 1
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espect to x 1 and (8.96) 2 with res
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For the two unknowns Xyy 1 and X 2x
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Redeveloping the determinant, the v
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291are obtained by equating to zero
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Let (z 0 , c 0 ) = (x 0 , y 0 , a 0
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we deduce that Γ 5 is submersive (
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11.4. Regularity and jet parametriz
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Cramer’s rule, we get(11.13) ⎧
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in K[a, z] n+m and in K[x, c] p+m .
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of the cancellation properties (11.
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305II: Explicit prolongations of in
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Define then the transformed jet ϕ
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Let λ ∈ N be an arbitrary intege
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311[Ol1986], [BK1989]):(Y(1.31)⎧
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+ [Y y 2 − 2 X xy ] y 1 y 2 + [
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for some nonnegative integers A, B,
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Page 319 and 320: By the classical formula for the de
Page 321 and 322: The induction formulas are⎧(3.4)
Page 323 and 324: We have to compute:( ) ⎛ ⎞∑
Page 325 and 326: Next, we gather the underlined term
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Page 329 and 330: Correspondingly, we identify the se
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Page 333 and 334: Y i1 ,i 2 ,i 3 ,i 4written in one o
Page 335 and 336: 335conclusion, we have shown that (
Page 337 and 338: 337Theorem 3.73. For every κ 1 an
Page 339 and 340: Theorem 3.79. For every integer κ
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Page 343 and 344: 343form:(4.12)κ+1Yκ j = Y jx κ +
Page 345 and 346: eing related to the number 2 in the
Page 347 and 348: Appying the chain rule, we may comp
Page 349 and 350: Secondly:(5.3)Y j i 1 ,i 2= Y jx i
Page 351 and 352: As explained before the statement o
Page 353 and 354: g j i 1 ,...,i λand similarly for
Page 355 and 356: with respect to the variables x i 1
Page 357 and 358: Assuming F = F(x, y x ) to be indep
Page 359 and 360: 359(III’) ⎧0 = ∑ (δ k 2)j 3H
Page 361 and 362: Open problem 1.17. For n = 2 establ
Page 363 and 364: ⎧∣ ⎨X 1 xX 1+ y x 1 y x 1 ·1
Page 365 and 366: eplacing the third column by the se
Page 367: ⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣
Page 371 and 372: Indeed, the collection of equations
Page 373 and 374: 3.12. Principal unknowns. As there
Page 375 and 376: 375Sixthly:(3.22) ⎧δ k 1j 1Θ n+
Page 377 and 378: are a consequence of (I’), (I”)
Page 379 and 380: 379IV: BibliographyREFERENCES[Ar198
Page 381 and 382: [G1989] GARDNER, R.B.: The method o
Page 383 and 384: [Me2005a] MERKER, J.: On the local
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