Since more than 4 months ago, I have posted a question on Mathstack and I haven't recieved any concrete answers. The link to the original post with the problem and my attempts are here.

To summarize, we need to prove that there exists a unique function which minimizes the seminorm of $H^1_0(\Omega)$ over the unit closed ball. We can use the approximation theorem for Hilbert spaces to minimize the norm. But does this necessarily minimize the seminorm? Otherwise is another approach better?

Any help is appreciated.