Given a convex set $O\subset \mathbb{R}^2$ with smooth boundary $\partial O$. Let $f\in C^2(\bar{O})\cap C^1(\mathbb{R}^2)$.
Question: Is it possible to find $g\in C^2(\mathbb{R}^2)$ such that $g=f$ on $O$ and $Dg(x)=Df(x)$ on $\partial O$?
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there are several questions here (with positve answers) which treat more general scenarios (e.g. the domain not being convex). The result is, in general, not easy to proof. It may be easier in your special case. If you are just interested in an answer, the following link may help: https://math.stackexchange.com/questions/131642/the-extension-of-smooth-function – Thomas Dec 29 '19 at 08:06