The Cauchy problem and Hadamard's example

Abstract

In 1917, Hadamard [5, p. 331 pointed out that U(X) = sin nx, sinh n.r, is a solution of the Laplace equation for every n and that this proves that the Cauchy problem for the Laplace equation with data on x1 = 0 is not well posed in the Cm-topology. Now we know that a necessary and sufficient condition for the Cauchy problem to be well posed in [w” is that the operator is hyperbolic in the Petrovskij-Girding sense in the direction normal to the initial hpperplane. For a presentation of Girding’s version of the theory see [6, Sects. 5.4 and 5.51. Later Larsson [S] settled the question in the case of nonanalytic Gevrey classes. In both cases, we get conditions on all of the terms of the operator, not just on the principal part. In the case of real analytic functions Bony and Schapira [2, 41 proved that the noncharacteristic Cauchy problem is always solvable in the class of real analytic functions in Iw” if the operator has a principal part which is hyperbolic in the direction normal to the initial hyperplane. Now we look at Hadamard’s example in a somewhat different form. The function U(X) = (1 xi f ix&l solves the Laplace equation outside m = (1, 0). It has the Cauchy data