smooth linear partial differential equation without solution

Cauchy-Kowalewski theorem says that real analytic partial differential equationsMathworldPlanetmath with real analytic initial data always have solutions. On the other hand Hans Lewy showed in 1957 that this is not true if the equation is only smooth. The example is obvious once we have the following theorem.

Theorem (Lewy).

Let x,y,z be independent real variables. Let f be a C1 real function. Suppose that there exists a C1 solution u to the following equation


in some neighbourhood of a point (0,0,z0). Then f is real analytic at z0.

Hence we need only pick f which is smooth and not real analytic at z0 and we have an example. For example, let z0=0 and f(x)=0xe-1/t𝑑t.


  • 1 Lewy, Hans. Ann. of Math. (2) 66 (1957), 155–158.
Title smooth linear partial differential equation without solution
Canonical name SmoothLinearPartialDifferentialEquationWithoutSolution
Date of creation 2013-03-22 17:39:38
Last modified on 2013-03-22 17:39:38
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 5
Author jirka (4157)
Entry type Example
Classification msc 35A10
Classification msc 35A05