Sharp Finiteness Principles for the boundary values of $C^2$-functions: long version

Abstract

Let $\Omega$ be a domain in ${\mathbf{R}}^n$ with boundary $\partial \Omega$. We prove the following Finiteness Principle for the boundary values of $C^2(\Omega )$-functions: A function $f:\partial \Omega \to \mathbf{R}$ is the trace to the boundary of a function $F\in C^2(\Omega )$ provided there exists a constant $\lambda >0$ such that for every set $E\subset \partial \Omega$ consisting of at most $N=3\cdot 2^{n-1}$ points, there exists a function $F_E\in C^2(\Omega )$ with $\Vert F_E{\Vert }_{C^2(\Omega )}\le \lambda$ whose trace to $\partial \Omega$ coincides with $f$ on $E$. Furthermore, we refine this principle by showing that this criterion can be restricted to $N$-point subsets $E\subset \partial \Omega$ that possess specific geometric ``visibility'' properties relative to $\Omega$.

93 pages, 11 figures