Holmgren's uniqueness theorem

In the theory of partial differential equations, Holmgren's uniqueness theorem, or simply Holmgren's theorem, named after the Swedish mathematician Erik Albert Holmgren (1873–1943), is a uniqueness result for linear partial differential equations with real analytic coefficients.

We will use the multi-index notation: Let ${\displaystyle \alpha =\{\alpha _{1},\dots ,\alpha _{n}\}\in \mathbb {N} _{0}^{n},}$, with ${\displaystyle \mathbb {N} _{0}}$ standing for the nonnegative integers; denote ${\displaystyle |\alpha |=\alpha _{1}+\cdots +\alpha _{n}}$ and

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