The Finiteness Principle for the boundary values of C²-functions

Let Ω be a domain in Rⁿ, and let N=3· 2ⁿ⁻¹. We prove that the trace of the space C²(Ω) to the boundary of Ω has the following finiteness property: A function f:∂Ω→ R is the trace to the boundary of a function F∈ C²(Ω) provided there exists a constant λ>0 such that for every set E⊂∂Ω consisting of at most N points there exists a function FE∈ C²(Ω) with \|FE\|C²(Ω)≤λ whose trace to ∂Ω 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⊂∂Ω which have some additional geometric ``visibility'' properties with respect to the domain Ω.