smooth linear partial differential equation without solution
Cauchy-Kowalewski theorem says that real analytic partial differential equations 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.
Let be independent real variables. Let be a real function. Suppose that there exists a solution to the following equation
in some neighbourhood of a point Then is real analytic at
Hence we need only pick which is smooth and not real analytic at and we have an example. For example, let and
- 1 Lewy, Hans. Ann. of Math. (2) 66 (1957), 155–158.
|Title||smooth linear partial differential equation without solution|
|Date of creation||2013-03-22 17:39:38|
|Last modified on||2013-03-22 17:39:38|
|Last modified by||jirka (4157)|