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smooth linear partial differential equation without solution
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(Example)
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Cauchy-Kowalewski theorem says that real analytic partial differential equations with real analytic initial data always have solutions. On the other hand Hans Lewy showed in 1957 that this is not true if the equation is only smooth. The example is obvious once we have the following theorem.
Theorem 1 (Lewy) Let $x,y,z$ be independent real variables. Let $f$ be a $C^1$ real function. Suppose that there exists a $C^1$ solution $u$ to the following equation \begin{equation*} \left[ - \frac{\partial}{\partial x} - i \frac{\partial}{\partial y} + 2i(x + i y) \frac{\partial}{\partial z} \right] u = f'(z) , \end{equation*}in some neighbourhood of a point $(0,0,z_0).$ Then $f$ is real analytic at $z_0.$
Hence we need only pick $f$ which is smooth and not real analytic at $z_0$ and we have an example. For example, let $z_0 = 0$ and $f(x) = \int_0^x e^{-1/t} ~ dt.$
- 1
- Lewy, Hans. An example of a smooth linear partial differential equation without solution. Ann. of Math. (2) 66 (1957), 155-158.
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"smooth linear partial differential equation without solution" is owned by jirka.
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Cross-references: point, neighbourhood, real function, variables, real, independent, theorem, obvious, smooth, equation, solutions, partial differential equations, real analytic, Cauchy-Kowalewski theorem
This is version 2 of smooth linear partial differential equation without solution, born on 2007-12-04, modified 2007-12-05.
Object id is 10094, canonical name is SmoothLinearPartialDifferentialEquationWithoutSolution.
Accessed 725 times total.
Classification:
| AMS MSC: | 35A05 (Partial differential equations :: General theory :: General existence and uniqueness theorems) | | | 35A10 (Partial differential equations :: General theory :: Cauchy-Kovalevskaya theorems) |
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Pending Errata and Addenda
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