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What do the symbols $C^{1,2}(\Sigma)$ and $C^{2,\alpha}(\Sigma)$ mean where $\Sigma$ is the domain? I know that $C^m(\Sigma)$ means $m$ times continuously differentiable functions. Also, what does $C^{\infty}_0(\Sigma)$ mean?

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    Are you studying *parabolic* differential equations? In this context one usually writes $C^{1, 2}$ to mean "$C^1$ with respect to the time variable, $C^2$ with respect to the space variables" (see e.g. Evans' book on PDEs).2013-09-02

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