[1] The Malgrange–Ehrenpreis theorem states (roughly) that linear partial differential equations with constant coefficients always have at least one solution; Lewy's example shows that this result cannot be extended to linear partial differential equations with polynomial coefficients.
using the following result: 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
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.