Given a domain $\Omega \subset \mathbb{R}^2$, one can reduce the Laplace equation $$\Delta u = 0, \qquad u = f \text{ on } \partial \Omega$$ to a Poisson equation $$\Delta v = g, \qquad v = 0 \text{ on } \partial \Omega$$ by finding some $C^2$ function $\phi$ on $\Omega$ whose boundary values equal $f$, and substituting $v = u-\phi$ and $g = -\Delta \phi$. The reason this might be desirable is that if one studies $\Delta$ as an operator in $L^2(\Omega)$, one needs a zero boundary condition in order for the operator to be self-adjoint.
My question is how to prove such a function $\phi$ exists (and if an explicit construction of $\phi$ is possible, so much the better). It's probably standard knowledge, but I haven't figured out where to look it up yet.
ADDENDUM: I'm looking for a relatively "elementary" approach to this extension problem (given $f$ on $\partial \Omega$, extend it to $\phi$ on $\Omega$), i.e. one that doesn't invoke the existence of solutions to the Dirichlet problem, the Riemann mapping theorem, etc. The reason I'm interested in the extension problem is that some of the numerous approaches to solving the Dirichlet problem appear to rely on it; as an example, chapter 7.2 of Lax's Functional Analysis discusses a method due to Garabedian and Schiffer in which one constructs a solution to the Dirichlet problem as a projection in a certain Hilbert space. Along the way he says [changing notation to fit mine] "We assume that $\partial \Omega$ is once differentiable, and that the boundary values of $u$ are also once differentiable. We can then construct a $C^1$ function $\phi$ on $\Omega \cup \partial \Omega$ that has the prescribed value on $\partial \Omega$." But no indication of how this extension $\phi$ is to be constructed. Is there some sort of $C^1$ version of Tietze extension/Urysohn's theorem?