This question come to me when I try to find the weak solution of following problem \begin{cases} -\Delta u =f&x\in\Omega \\\ u=g&x\in\partial\Omega \end{cases} where $\Omega$ is open bounded smooth boundary.
Then the usual trick we always take is to find a function $\tilde g\in H^2(\Omega)$ such that $T[\tilde g]=g$ and we study the problem \begin{cases} -\Delta \tilde u =f-\Delta \tilde g&x\in\Omega \\\ \tilde u=0&x\in\partial\Omega \end{cases} and hence we could apply Lax-Milgram and so on.
However, I want to ask that can we always find such $\tilde g$? That is, given any function $g\in L^2(\partial \Omega)$, is there always exists a function $\tilde g\in H^2(\Omega)$ such that $T[\tilde g]=g$? i.e., is that the trace operator $T$ is invertible? We know $T$: $H^2(\Omega)\to L^2(\partial\Omega)$ (actually is $H^1(\Omega)\to L^2(\partial\Omega)$) is well defined, but how about $T^{-1}$? Or is that $T$ is surjective?