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# Holmgren uniqueness theorem

Given a system of linear partial differential equations with analytic coefficients $a_{{ji_{1},\ldots,i_{m}}}$ and $b_{i}$

$\sum_{{j,i_{1},\ldots,i_{m}}}a_{{ji_{1},\ldots,i_{m}}}(x_{1},\ldots,x_{m}){% \partial^{{i_{1}+\cdots+i_{m}}}u_{j}\over\partial x_{1}^{{i_{1}}}\ldots% \partial x_{m}^{{i_{m}}}}=b_{i}(x_{1},\ldots,x_{m})$ |

and analytic Cauchy data specified along a noncharacteristic analytic surface, there exists a neighborhood of the surface such that every smooth solution of the system defined in that neighborhood is analytic.

This theorem stengthens the Cauchy-Kowalewski theorem. While the latter theorem asserts that a unique analytic solution exists, it still allows the possibility that there might exist non-analytic solutions. Holmgren’s theorem asserts that this is not the case for linear systems.

Type of Math Object:

Theorem

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## Mathematics Subject Classification

35A10*no label found*

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