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

370Of course, to define the square functions in the context of n 2 independentvariables (x 1 , x 2 , . . ., x n ), we introduce the Jacobian determinantX 1 x· · · X 1 ∣ 1 x n X1 y ∣∣∣∣∣∣∣ (2.36) ∆(x 1 |x 2 | · · · |x n |y) :=. · · · . .,X n x· · · X n∣1 x n Xn yY x 1 · · · Y x n Y ytogether with its modifications(2.37) ∆ ( x 1 | · · · | k 1x j 1x j 2| · · · |y ) ,in which the k 1 -th column of partial first order derivatives | k 1x k 1| is replacedby the column | k 1x j 1x j 2| of partial derivatives. Here, the indices k 1 , j 1 ,j 2 satisfy 1 k 1 , j 1 , j 2 n + 1, with the convention that we adopt thenotational equivalence(2.38) x n+1 ≡ y .This convention will be convenient to write some of our general formulas inthe sequel.As we promised to only summarize the proof of Theorem 1.7 in this paper,we will not develope the proof of Lemma 2.33: it is similar to the proof ofLemma 3.32 in [Me2004].§3. FIRST AND SECOND AUXILIARY SYSTEM3.1. Functions G j1 ,j 2, H k 1j 1 ,j 2, L k 1j 1and M k 1. To discover the four families offunctions appearing in the statement of Theorem 1.7, by comparing (2.35)and (1.10), it suffices (of course) to set:(3.2)⎧⎪⎨⎪⎩G j1 ,j 2:= −□ n+1x j 1x , j 2H k 1j 1 ,j 2:= □ k 1x j 1x − j 2 δk 1j 1□ n+1x j 2y − δk 1j 2□ n+1x j 1y ,L k 1j 1:= 2 □ k 1x j 1y − δk 1j 1□ n+1yy ,M k 1:= □ k 1yy.Consequently, we have shown the “only if” part of Theorem 1.7, which isthe easiest implication.To establish the “if” part, by far the most difficult implication, the verymain lemma can be stated as follows.Lemma 3.3. The partial differerential relations (I’), (II’), (III’) and (IV’)which express in length the compatibility conditions (1.11) are necessaryand sufficient for the existence of functions X l , Y of (x l 1, y) satisfying thesecond order nonlinear system of partial differential equations (3.2) above.