# Lewy's example

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{{Short description|Linear partial differential equation with no solutions}}
In the [mathematical](/source/Mathematics) study of [partial differential equation](/source/partial_differential_equation)s, '''Lewy's example''' is a celebrated example, due to [Hans Lewy](/source/Hans_Lewy), of a [linear partial differential equation](/source/Partial_differential_equation) with no solutions. It shows that the analog of the [Cauchy–Kovalevskaya theorem](/source/Cauchy%E2%80%93Kovalevskaya_theorem) does not hold in the smooth category.

The original example is not explicit, since it employs the [Hahn–Banach theorem](/source/Hahn%E2%80%93Banach_theorem), but there since have been various explicit examples of the same nature found by [Howard Jacobowitz](/source/Howard_Jacobowitz).<ref>{{Citation |last=Jacobowitz |first=Howard |title=Systems of homogeneous partial differential equations with few solutions |date=1988 |url=http://link.springer.com/10.1007/BFb0100788 |work=Partial Differential Equations |volume=1324 |pages=127–136 |editor-last=Cardoso |editor-first=Fernando |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |doi=10.1007/bfb0100788 |isbn=978-3-540-50111-4 |editor2-last=de Figueiredo |editor2-first=Djairo G. |editor3-last=Iório |editor3-first=Rafael |editor4-last=Lopes |editor4-first=Orlando|url-access=subscription }}</ref>

The [Malgrange–Ehrenpreis theorem](/source/Malgrange%E2%80%93Ehrenpreis_theorem) states (roughly) that linear partial differential equations with constant [coefficient](/source/coefficient)s always have at least one solution; Lewy's example shows that this result cannot be extended to linear partial differential equations with [polynomial](/source/polynomial) coefficients.

==The example==

The statement is as follows
:On <math>\mathbb{R} \times \mathbb{C}</math>, there exists a [smooth](/source/smooth_function) (i.e., <math>C^{\infty}</math>) [complex](/source/complex_number)-valued [function](/source/function_(mathematics)) <math>F(t,z)</math> such that the [differential equation](/source/differential_equation)
::<math>\frac{\partial u}{\partial\bar{z}}-iz\frac{\partial u}{\partial t} = F(t,z)</math>
:admits no solution on any [open set](/source/open_set). Note that if ''<math>F</math>'' is [analytic](/source/analytic_function) then the [Cauchy–Kovalevskaya theorem](/source/Cauchy%E2%80%93Kovalevskaya_theorem) implies there exists a solution.

Lewy constructs this ''<math>F</math>'' using the following result:
:On <math>\mathbb{R} \times \mathbb{C}</math>, suppose that <math>u(t,z)</math> is a function satisfying, in a [neighborhood](/source/neighborhood_(mathematics)) of the [origin](/source/origin_(mathematics)),
::<math>\frac{\partial u}{\partial\bar{z}}-iz\frac{\partial u}{\partial t} = \varphi^\prime(t) </math>
:for some ''C''<sup>1</sup> function ''&phi;''.  Then ''&phi;'' must be [real-analytic](/source/real-analytic) in a (possibly smaller) neighborhood of the origin.

This may be construed as a non-existence theorem by taking ''&phi;'' 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 <math>\mathbb{R} \times \mathbb{C}</math>.  The method of [proof](/source/mathematical_proof) uses a [Baire category](/source/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
:<math>\frac{\partial u}{\partial x}+ix\frac{\partial u}{\partial y} = F(x,y)</math>
depending on 2 [real](/source/real_number) variables ''x'' and ''y'' sometimes has no solutions. This is almost the simplest possible [partial differential operator](/source/Differential_operator) with non-constant coefficients.

==Significance for CR manifolds==
A [CR manifold](/source/CR_manifold) comes equipped with a [chain complex](/source/chain_complex) of differential operators, formally similar to the [Dolbeault complex](/source/Dolbeault_complex) on a [complex manifold](/source/complex_manifold), called the <math>\scriptstyle\bar{\partial}_b</math>-complex.  The Dolbeault complex admits a version of the [Poincaré lemma](/source/Poincar%C3%A9_lemma).  In the language of [sheaves](/source/sheaf_(mathematics)), this means that the Dolbeault complex is exact.  The Lewy example, however, shows that the <math>\scriptstyle\bar{\partial}_b</math>-complex is almost never exact.

== Notes ==
{{Reflist}}

==References==
*{{citation
| last = Lewy
| first = Hans
| author-link = Hans Lewy
| title = An example of a smooth linear partial differential equation without solution 
| journal = [Annals of Mathematics](/source/Annals_of_Mathematics)
| volume = 66
| issue = 1
| year = 1957
| pages = 155–158
| jstor = 1970121
| mr = 0088629
| zbl = 0078.08104 
| doi = 10.2307/1970121
}}.
*{{citation
| first = Sigeru
| last = Mizohata
| author-link = Sigeru Mizohata
| title = Solutions nulles et solutions non analytiques
| journal = Journal of Mathematics of Kyoto University
| volume = 1  
| issue = 2
| year = 1962 
| language = French
| url = http://projecteuclid.org/euclid.kjm/1250525061
| pages= 271–302
| mr = 142873
| zbl = 0106.29601
| doi = 
}}.
*{{springer|id=l/l120080|title=Lewy operator and Mizohata operator|first=Jean-Pierre |last=Rosay}}

Category:Partial differential equations

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