A Mini-course on Elliptic–Hyperbolic Equations - ICMS

Writing the characteristic equation for Lu = 0, we haveKdη 2 + γdξ 2 = 0. (4)If K and K y can vanish simultaneously, then it is possible to haveγ ≠ 0 on the sonic curve, in which case eq. (4) assumes the formγdξ 2 = 0, γ ≠ 0. So in the case K = 0 = K y the characteristiclines satisfy ξ = dξ = 0. That is, they degenerate on the soniccurve, and the case K = 0 = K y corresponds to equations ofKeldysh type.A tedious algebraic calculation will show directly that if K and K ydo not vanish simultaneously, then the operator L is of Tricomitype.That is to say, the Tricomi operator (K (x, y) = y) is of realprincipal type but the Keldysh operator (K (x, y) = x) is not, anda lower-order perturbation of a well-behaved operator of Keldyshtype can be badly behaved.