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| In the [[Mathematics|mathematical]] study of [[partial differential equation]]s, '''Lewy's example''' is a celebrated example, due to [[Hans Lewy]], of a linear partial differential equation with no solutions. It shows that the analog of the [[Cauchy–Kovalevskaya theorem]] does not hold in the smooth category.
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| The original example 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]].
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| The [[Malgrange–Ehrenpreis theorem]] states (roughly) that linear partial differential equations with [[constant 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 coefficients.
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| ==The Example==
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| The statement is as follows
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| :On ℝ×ℂ, there exists a [[Smooth function|smooth]] complex-valued function <math>F(t,z)</math> such that the differential equation
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| ::<math>\frac{\partial u}{\partial\bar{z}}-iz\frac{\partial u}{\partial t} = F(t,z)</math>
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| :admits no solution on any open set. Note that if ''<math>F</math>'' is analytic then the [[Cauchy–Kovalevskaya theorem]] implies there exists a solution.
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| Lewy constructs this ''<math>F</math>'' using the following result:
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| :On ℝ×ℂ, suppose that <math>u(t,z)</math> is a function satisfying, in a neighborhood of the origin,
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| ::<math>\frac{\partial u}{\partial\bar{z}}-iz\frac{\partial u}{\partial t} = \varphi^\prime(t) </math>
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| :for some ''C''<sup>1</sup> function ''φ''. Then ''φ'' must be real-analytic in a (possibly smaller) neighborhood of the origin.
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| 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.
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| {{harvtxt|Mizohata|1962}} later found that the even simpler equation
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| :<math>\frac{\partial u}{\partial x}+ix\frac{\partial u}{\partial y} = F(x,y)</math>
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| 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.
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| ==Significance for CR manifolds==
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| A [[CR manifold]] comes equipped with a [[chain complex]] of differential operators, formally similar to the [[Dolbeault complex]] on a [[complex manifold]], called the <math>\scriptstyle\bar{\partial}_b</math>-complex. The Dolbeault complex admits a version of the [[Poincaré lemma]]. In the language of [[sheaf (mathematics)|sheaves]], 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.
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| ==References==
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| *{{citation
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| | last = Lewy
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| | first = Hans
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| | author-link = Hans Lewy
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| | title = An example of a smooth linear partial differential equation without solution
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| | journal = [[Annals of Mathematics]]
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| | volume = 66
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| | issue = 1
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| | year = 1957
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| | pages = 155–158
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| | jstor = 1970121
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| | mr = 0088629
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| | zbl = 0078.08104
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| | doi = 10.2307/1970121
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| }}.
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| *{{citation
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| | first = Sigeru
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| | last = Mizohata
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| | author-link = Sigeru Mizohata
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| | title = Solutions nulles et solutions non analytiques
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| | journal = Journal of Mathematics of Kyoto University
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| | volume = 1
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| | issue = 2
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| | year = 1962
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| | language = [[French language|French]]
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| | url = http://projecteuclid.org/euclid.kjm/1250525061
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| | pages= 271–302
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| | mr = 142873
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| | zbl = 0106.29601
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| | doi =
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| }}.
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| *{{springer|id=l/l120080|title=Lewy operator and Mizohata operator|first=Jean-Pierre |last=Rosay}}
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| [[Category:Partial differential equations]]
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