In lecture we had the following theorem:
Let $\Omega \subseteq \subseteq \mathbb{R}^n$. Then $$H^1(\Omega) = H^1_0(\Omega) \oplus \{u \in H^1 : \Delta u = 0\}$$ where $\Delta u$ is understood in the distributional sense. Moreover, if $\Omega$ is of class $C^1$, $u\vert_{\partial \Omega}$ does admit a unique harmonic extension.
Now the proof of the above is straight forward, however, the moreover part is where I stumble: In the notes it is phrased that this is a direct consequence of the decomposition, but I do not see this. I mean, we do have $$u\vert_{\partial \Omega} = u_0\vert_{\partial\Omega} + u_1 \vert_{\partial \Omega} = u_1\vert_{\partial \Omega}$$ when $u = u_0 + u_1$ and hence $u_1$ is a harmonic extension of $u\vert_{\partial \Omega}$, but why is this extension unique? The reason why I am asking this is because I need to proof that $$H^1_0(\Omega) = \{u \in H^1(\Omega) : u\vert_{\partial \Omega} = 0\}.$$