Travaux sur les symétries de Lie des équations aux ... - DMA - Ens

231for (7.28):( )∂□ ·∂a □ν µ [01+l2 ] − □ν[01+l2 ] · ( )∂∂a □ = µ= −□ ( 0 [1+l2 ]| · · · | ν a ν | · · · |a n+1) · □ ( a 1 | · · · | ν a 1 a µ | · · · |a n+1) − · · · −− □ ( a 1 | · · · | ν 0 [1+l2 ]| · · · |a n+1) · □ ( a 1 | · · · | ν a ν a µ | · · · |a n+1) − · · · −− □ ( a 1 | · · · | ν a ν | · · · |0 [1+l2 ])· □(a 1 | · · · | ν a n+1 a µ | · · · |a n+1) ,or equivalently, in contracted form:( )∂□ ·∂a □ν µ [01+l2 ] − □ν[01+l2 ] · ( ) ∑n+1∂∂a □ = − µτ=1□ τ [0 1+l2 ] · □ν [a τ a µ ] .∂Thanks to this sidework, coming back to the expression for2 Gwe left∂y xl 1∂y xl 2pending above, we obtain:∂ 2 G= 1 ∑n+1∂y x l 1 ∂y x l 2 □ 2µ=1∑n+1ν=1− 1 ∑n+1□ 3µ=1{} ∂□ µ 2 T[0 1+l1 ]· □ν [0 1+l2 ]∂a µ ∂a − ν∑n+1ν=1∑n+1τ=1{} ∂T□ µ [0 1+l1 ] · □τ [0 1+l2 ] · □ ν [a µ a τ ]∂a . νTo really finalize this expression, we factor everything by1 and we exchangethe two summation indices ν and τ in the second line:□ 3{}∂ 2 G= 1 ∑n+1∑n+1□ µ ∂ 2 n+1T∂y x l 1 ∂y x l 2 □ 3 [0 1+l1 ]· □ν [0 1+l2 ] □ ·∂a µ ∂a − ∑□ τ ν [a µ a ν ] · ∂T∂a τµ=1ν=1τ=1.End of proof of the Main Theorem. As already explained in the Introduction,one applies to the system (7.28) Hachtroudi’s characterization (7.28)of equivalence to the system w ′ z ′ k 1z ′ k 2(z ′ ) = 0 with x := z, with y := w, witha := z, with b := w, with (a, b) := t, with Q := Θ, with □ := ∆, withG := Φ k1 ,k 2and with T :=∂2 Θ1∂z k1 ∂z k2. The denominator can be cleared∆ 3out, and we simply get the explicit fourth-order partial differential equationsatisfied by Θ. This completes the proof of our Main Theorem and the papermay end up now.REFERENCES[1] Bièche, C.: Le problème d’équivalence locale pour un système scalaire complet d’équationsaux dérivées partielles d’ordre deux à n variables indépendantes, Annales de la Faculté desSciences de Toulouse, XVI (2007), no. 1, 1–36.[2] Boggess, A.: CR manifolds and the tangential Cauchy-Riemann complex. Studies in AdvancedMathematics. CRC Press, Boca Raton, FL, 1991, xviii+364 pp.[3] Cartan, É.: Sur les variétés à connexion projective, Bull. Soc. Math. France 52 (1924), 205–241.