Lewy's example

In mathematics, in the field partial differential equations, Lewy's example is a celebrated example, due to Hans Lewy, of a linear partial differential equation with no solutions. It removes the hope that the analog of the Cauchy-Kovalevskaya theorem can hold, in the smooth category.

The "example" itself is not explicit, since it employs the Hahn-Banach theorem, but there since have been various explicit examples of the same nature found by Harold Jacobowitz.

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.

The Example

The statement is as follows:On RxC, there exists a smooth complex-valued function "F"("t","z") such that the differential equation:::admits no solution on any open set. (If "F" is analytic then the Cauchy-Kovalevskaya theorem implies there is a solution.)

Lewy constructs this "F" using the following result::On RxC, suppose that "u"("t","z") is a function satisfying, in a neighborhood of the origin,:::for some "C"¹ 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 RxC. The method of proof uses a Baire category argument, so in a certain precise sense almost all equations of this form are unsolvable.

harvtxt|Mizohata|1962 later found that the even simpler equation:$frac\{partial\; u\}\{partial\; x\}+ixfrac\{partial\; u\}\{partial\; y\}\; =\; F(x,y)$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.

ignificance for CR manifolds

A CR manifold comes equipped with a chain complex of differential operators, formally similar to the Dolbeault complex on a complex manifold, called the -complex. The Dolbeault complex admits a version of the Poincaré lemma. In the language of sheaves, this asserts that the Dolbeault complex is exact. The Lewy example, however, shows that the -complex is almost never exact.

References

*citation|id=MR|0088629|last= Lewy|first= H|title=An example of a smooth linear partial differential equation without solution |journal=Ann. of Math.|volume=66|year=1957|pages= 155-158
url=http://links.jstor.org/sici?sici=0003-486X%28195707%292%3A66%3A1%3C155%3AAEOASL%3E2.0.CO%3B2-X
*
*springer|id=l/l120080|title=Lewy operator and Mizohata operator|first=Jean-Pierre |last=Rosay