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When the weak derivative just is the strong (or classical) derivative? For instance, can we prove that weak derivate $Du\in C^\alpha$(or $C^0$) implies $u\in C^{1,\alpha}$(or $C^1$).

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    When it comes to equivalence classes up to set of zero measure, you cannot hope to prove esily pointwise results. There is a nice paragraph about weak and pointwise derivatives in these notes: http://www.math.ucsd.edu/~bdriver/231-02-03/Lecture_Notes/weak-derivatives.pdf2012-11-26

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