Any connection between the trace operator and the trace of a matrix?

Question

We have the concept of trace operator in functional analysis and PDE theory at wiki. My first question is, as the informal discussion stated, if $u$ is a function that is $C^1$ (or even just continuous) on the closure $\bar \Omega$ of $\Omega$, i.e., $u \in C^1(\bar\Omega)$, its function restriction, say $u|_{\partial \Omega}$, is well-defined and continuous on $\partial \Omega$. Then it noted: "Remember that $\Omega$ is an open set, so boundary points are not in $\Omega$ and consequently $u$ itself is not defined at boundary points." I don't understand why $u$ is undefined at $\partial \Omega$ if $u \in C^1(\bar\Omega)$.

My second question is about the idea of trace operator linked to the trace of matrix. I first learned the term of trace from linear algebra that is defined for the sum of diagonal elements of a matrix. Is there any connection between the trace of a matrix and the trace operator of a function in Sobolev space?

Answer 1

Before that sentence, the article discusses functions $u:\Omega\to \mathbb R$, and says the trace is the limit $\lim_{x\to x_0, \ x\in \Omega} u(x)$ with $u_0\in \partial \Omega$ (if it exists). If this limit exists then everything is fine (i.e., $u$ continuous from the closure of $\Omega$ to $\mathbb R$, or $u$ uniformly continuous from $\Omega$ to $\mathbb R$).

There is no relation between the two notions of traces.