The Finiteness Principle for the boundary values of $C^2$-functions

Let $\mathrm{\Omega }$ be a domain in $R^n$, and let $N=3\cdot 2^{n-1}$. We prove that the trace of the space $C^2(\mathrm{\Omega })$ to the boundary of $\mathrm{\Omega }$ has the following finiteness property: A function $f:\mathrm{\partial }\mathrm{\Omega }\to R$ is the trace to the boundary of a function $F\in C^2(\mathrm{\Omega })$ provided there exists a constant $\lambda >0$ such that for every set $E\subset \mathrm{\partial }\mathrm{\Omega }$ consisting of at most $N$ points there exists a function $F_E\in C^2(\mathrm{\Omega })$ with $\mathrm{\parallel }{F}_E{\mathrm{\parallel }}_{C^2(\mathrm{\Omega })}\le \lambda$ whose trace to $\mathrm{\partial }\mathrm{\Omega }$ coincides with $f$ on $E$. We also prove a refinement of this finiteness principle, which shows that in this criterion we can use only $N$-point subsets $E\subset \mathrm{\partial }\mathrm{\Omega }$ which have some additional geometric ``visibility’’ properties with respect to the domain $\mathrm{\Omega }$.