I want to prove that for open region $\Omega\subset \mathbb{R}^d$ with certain boundary regularity and continuous function $f$, the Dirichlet problem$$\Delta u = 0, \quad u|_{\partial \Omega} = f$$has a solution. I saw some proofs before, and I came up with the following proof that seems simpler than the common methods, so I am not sure whether it is correct. As suggested by Chapter 5 Lemma 4.10 in Stein's Real Analysis, we only need to consider the case that $f$ is the boundary value given by some $F\in C^2(\bar{\Omega})$ since general continuous $f$ can be uniformly approximate by smooth functions. (It was $C^1$ in that original lemma, but I think this does not matter much.)
Now suppose $f = F|_{\partial \Omega}$ such that $F\in C^2(\bar{\Omega})$, consider $v = u-F$, then we only need to find a solution for\begin{equation}\Delta v = -\Delta F,\quad v_{\partial \Omega} = 0.\end{equation}Consider its weak form$$\int_{\Omega}\nabla v\nabla \phi = \int_{\Omega} \phi\Delta F$$where $v$ and the test function $\phi$ are all in Hilbert space $H_0^1(\Omega)$. By Poincaré inequality, we know the bilinear form $B(v,\phi) := \int_{\Omega}\nabla v\nabla \phi$ is coercive, and bounded under $H^1$ norm. Now we may use the Lax-Milgram lemma to deduce that the equation of $v$ has a weak solution. After substituting back $u = v+F$, we get a weak solution for the original Dirichlet problem, in the sense that$$\int_{\Omega} u\Delta \phi = 0,$$at least for sufficiently smooth $\phi$. Finally, we may use the fact that a weak harmonic function agrees with a harmonic function almost everywhere. (This is established by proving the mean value property for weak harmonic functions, and one can see details in Chapter 5 Theorem 4.3 in Stein's Real Analysis).
It seems like a relatively simple proof for the Dirichlet principle, but I am skeptical about its correctness since it does not use the regularity for $\partial \Omega$ at all. However, some references point out that the regularity for $\partial \Omega$ is crucial for this problem, for example, this article as well as Real Analysis.