A Mini-course on Elliptic–Hyperbolic Equations - ICMS

Alternative definition of the canonical forms:We can always write a second-order linear equation with smoothcoefficients defined in an open set of R 2 in the local formLu = K (x, y) u xx + u yy + lower-order terms.One can, by a further nonsingular transformation, arrive at acoordinate system in which the type-change function K (x, y) haseither of two forms:1. K(y), with K(0) = 0 and yK(y) > 0 for y ≠ 0, or2. K(x), with K(0) = 0 and xK(x) > 0 for x ≠ 0.The former characterizes equations of Tricomi type and the latter,equations of Keldysh type.Still more generally, one notices that the distinction is onlymeaningful in a neighborhood of the sonic curve.