Lewy's example

The example of Lewy is an example of a partial differential equation with no smooth solutions, even though all the data of the equation are smooth.

Lange had been supposed to be able to build an analog to the theory of ordinary differential equations, existence and uniqueness theory, at least for linear partial differential equations. The Cauchy - Kovalevskaya (1875 ) seemed to point the way: Each correct Asked Cauchy problem with analytical data has an analytical solution. Since the beginning of the 20th century, many partial differential equations were solved and the experience showed that Differenzierbarkeiteigenschaften the data of the equation lead to possibly influenced by the degree equation differentiability of the solutions. It was therefore assumed suggest that a set of Cauchy- Kovalevskaya to the analogous statement is true when going from analytic functions to smooth functions. The surprisingly simple example of Lewy disproved this conjecture and Hans Lewy himself writes:

The example of Lewy is a linear partial differential equation of first order for complex-valued functions in three unknowns:

In a first step Lewy showed that when the right side is equal to one only of dependent and once continuously differentiable function and if there is in a neighborhood of a once continuously differentiable solution, then in must be analytical. Lewy used this to find using Banach space arguments on non- constructive way, a smooth function, so that the above equation has no solution, the latter being the space of all functions is their first partial derivatives exist and a Hölder condition suffice for all pairs of points with distance. In particular, the linear partial differential equation has no smooth solution with this as the right side.

This example is of first order and only the coefficient before the derivative with respect to is non- constant, but as a polynomial ( even the first degree) very easy. Therefore, the example of Lewy also shows that it is not possible to generalize in a simple way, the set of Malgrange - Ehrenpreis.