Here is what I know. Let $\Omega \subset \mathbb{R}^n$ be bounded and $\partial \Omega$ be $C^1$. If $f \in C(\Omega)$ then there, if $T$ is the classical trace operator, $Tf = f|_{\partial \Omega}$ is well defined. However, for $f \in W{1,p}(\Omega)$, $Tf$ is not well defined because $f$ is only defined a.e. and $\partial \Omega$ has Lebesgue measure zero. Since $T$ is a bounded linear operator and since $C^\infty$ functions are dense in $L^p$, we may extend $T$ to now be defined as $$T: W^{1,p}(\Omega) \rightarrow L^p(\partial \Omega).$$ Thus we have a well defined notion for the boundary of a $W^{1,p}$ function.
I would like to know how this theorem is used in practice. For example, suppose $L$ is a differential operator that gives the following PDE: $$\begin{cases} Lu = 0, \quad x \in \Omega \\ u = g, \quad x \in \partial \Omega \end{cases}$$
The theorem does not explicitly tell me what the image of $Tu$ is, only that it exists. How can I use the trace theorem to ensure the boundary value of $u$ is indeed $g$ and how is this often done in practice? Is this usually done by some sort of density argument?
The trace operator has a pretty nice characterisation as follows (see e.g. "Partial Differential Equations" by L.C. Evans, Chapter 5.5): If $\Omega$ is bounded and $\partial \Omega$ is $C^1$ then we have for $u \in W^{1,p}(\Omega)$ that $u \in W_0^{1,p}(\Omega)$ if and only if $Tu=0$ ond $\partial U$.
Here $W_0^{1,p}(\Omega)$ is the closure of $C_0^\infty(\Omega)$ in the $W^{1,p}$-norm.
We therefore transform the boundary condition as follows: Since $u,g$ are not continuous their boundary value is not defined in the usual sense. We therefore interpret $u=g$ on $\partial \Omega$ as $Tu=Tg$ on $\partial \Omega$. The trace operator is linear, so this is equivalent to $T(u-g)=0$. With the given characterisation of function with trace zero this is equivalent to $u-g \in W_0^{1,p}(\Omega)$ or $u \in g+ W_0^{1,p}(\Omega)$.
The last two conditions are now our boundary condition. We therefore now look for a function $u$ such that $Lu=0$ on $\Omega$ (probably in the weak sense) and $u-g \in W_0^{1,p}(\Omega)$. The last property is equivalent to $u-g$ being approximated in the $W^{1,p}$-norm by $C_0^\infty(\Omega)$ functions (smooth functions with compact support). In other words we only look for solutions $u$ that can be written as the limit of a sequence $(g+u_n)_n$ with $u_n \in C_0^\infty(\Omega)$ for all $n$.