This is a close relative of the following problem.

Let $\Omega$ be an open, bounded subdomain of $\mathbb R^n$ with smooth boundary, and $f_i \in W^{1, \infty} (\Omega)$ a sequence of functions equibounded in $W^{1, \infty}$ norm - that is, there exists some $M > 0$ such that $\|f_i\|_{W^{1, \infty} (\Omega)} < M$ for each $i$.

Define the functional $L: W^{1, \infty} (\Omega) \to \mathbb R$ by

$$L(h) = \limsup_{i \to \infty} \|f_i - h\|_{L^\infty (\Omega)},$$

and write

$$L_{|U} (h) := \limsup_{i \to \infty} \|f_i - h\|_{L^\infty (U)}$$

for the restriction of the functional $L$ to an open set $U \subset \mathbb R^n$.

We say $f \in W^{1, \infty} (\Omega)$ is an absolute minimizer of $L$ if for every open ball $B$, and every $g \in W^{1, \infty}(\Omega)$ such that $f = g$ on $\partial B$, we have

$$L_{|B} (f) \leq L_{|B} (g)$$

Question: Does the functional $L$ admit an absolute minimizer?