Why minimize $ \Vert u_1 + u \Vert_1 $ in this Finite Element Analysis variational problem? $\textbf{The problem reads:}$
Let $\Omega$ be bounded with $\Gamma: = \partial \Omega$ and let $g:\Gamma \rightarrow \mathbb{R} $ be a given function. Find the function $u\in H^1(\Omega)$ with minimal $H^1$- norm which coincides with $g$ on $\Gamma$. 
$\textbf{The confusion:}$
We have to minimise $ \Vert u_1 + u \Vert_1 $. 
Why? The plus is strange as we are not minimising a difference.
$\textbf{The attempt:}$
Let $g$ be a restriction on the function $u_1 \in C(\bar{\Omega})$. We look thus for $u \in H^1_0(\Omega)$ as the following transform creates homogeneous boundary conditions:
$$w = 0 = u - u_1 \quad \text{on  } \partial \Omega$$
How does one get to minimizing the above mentioned from this?
Would you please help guide me in this. I'm a bit lost as one can see. Thank you.
Note: $\Vert \cdot \Vert_1 $ is the $H^1$- norm

If I'm understanding correctly, it sounds like the intent is to solve the problem
$$\arg \min_{u \in H^1,u|_\Gamma=g} \| u \|_{H^1}$$
by fixing some $u_1 \in H^1$ with $u_1|_\Gamma=g$ and then solving
$$\arg \min_{u \in H^1_0} \| u_1+u \|_{H^1}.$$
Now presumably you go from the variational problem to an inhomogeneous linear PDE with homogeneous boundary conditions, where the forcing term originates in this $u_1$ that you selected.
