Minimum solution over closed ball of $H_0^1(\Omega)$

Question

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.

Answer 1

I think there's a mistake in the definition of your norm $\|\cdot\|_\kappa$ : it does not seem to be equivalent to the $H^1(\Omega)$ norm (since there's no gradients involved).

I would more simply consider the following norm on $H^1_0(\Omega)$ : \begin{align*} N(v) := \left(\int_\Omega \kappa |\nabla v|^2\right)^{1/2}. \end{align*} Since $0<\inf \kappa \leq \sup \kappa <+\infty$ and $\Omega$ is bounded, Poincaré's inequality ensures you that $N$ is equivalent to the usual $H^1(\Omega)$ norm, so you have completeness and $S$ remains closed. Better : you still have the hilbertian structure, since \begin{align*} \langle u,v\rangle_N := \int_{\Omega} \kappa \nabla v \cdot\nabla u, \end{align*} defines an inner-product. Minimizing $N(u-v)$ for $v\in S$ solves your question.