Lewy's Example - The Example

The Example

The statement is as follows

On ℝ×ℂ, there exists a smooth complex-valued function such that the differential equation

admits no solution on any open set. Note that if is analytic then the Cauchy–Kovalevskaya theorem implies there exists a solution.

Lewy constructs this using the following result:

On ℝ×ℂ, suppose that is a function satisfying, in a neighborhood of the origin,

for some C1 function φ. Then φ must be real-analytic in a (possibly smaller) neighborhood of the origin.

This may be construed as a non-existence theorem by taking φ to be merely a smooth function. Lewy's example takes this latter equation and in a sense translates its non-solvability to every point of ℝ×ℂ. The method of proof uses a Baire category argument, so in a certain precise sense almost all equations of this form are unsolvable.

Mizohata (1962) later found that the even simpler equation

depending on 2 real variables x and y sometimes has no solutions. This is almost the simplest possible partial differential operator with non-constant coefficients.