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

TL;DR

This paper establishes a finiteness principle for the boundary values of twice continuously differentiable functions, showing that boundary data can be characterized by finite, geometrically visible subsets of the boundary.

Contribution

It introduces a new finiteness criterion for boundary traces of $C^2$ functions, involving only finite subsets with specific geometric properties.

Findings

Finiteness property for boundary traces of $C^2$ functions.

Existence of $C^2$ extensions from finite boundary data.

Refinement involving geometrically visible boundary subsets.

Abstract

Let $\Omega$ be a domain in $R^n$, and let $N=3\cdot 2^{n-1}$. We prove that the trace of the space $C^2(\Omega)$ to the boundary of $\Omega$ has the following finiteness property: A function $f:\partial\Omega\to R$ 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 $N$ points there exists a function $F_E\in C^2(\Omega)$ with $\|F_E\|_{C^2(\Omega)}\le\lambda$ whose trace to $\partial\Omega$ 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 $E\subset\partial\Omega$ which have some additional geometric ``visibility'' properties with respect to the domain $\Omega$.