Since I am from a different mathematical field and couldn't find it: Is there something which would be best called an Arzela-Ascoli version for the $L_p$-norm, namely:

Let $X,Y$ be two nice measurable/normed spaces (i.e. compact, locally compact, hausdorff and whatever properties you might want), e.g. $X$ a closed interval and $Y= \mathbb R$.

Let now $f_n$ be a family of continuous maps from $X$ to $Y$ such that their $L_p$-norm converges, i.e. $||f_n||_p \to \lambda$. Does there exist a converging subfamily $f_k \to f$ such that $f$ is continuous and $||f||_p=\lambda$.

The question arose from the following setting: Let $X$ and $Y$ be two finite metric graphs (you can think of them as closed intervals glued together at their endpoints with metric & measure from $\mathbb R$). Given a family of piecewise differentiable maps $f_n: X \to Y$, does there exist a piecewise differentiable map $f: X \to Y$, such that the $L_p$-norm of its derivative $||f'||_p$ attains the infimum of $||f'_n||_p$?

Here the family $f_n$ is the set of all piecewise differentiable maps homotopic to a given function. Hence I think we can even assume that the $f_n$ are continuous and piecewise linear, since averaging along edges decreases the $L_p$-norm of its derivative , i.e. do we have $||f'_n||_p \geq |\frac{f_n(b)-f_n(a)}{b-a}|$ for $f_n:[a,b] \to \mathbb R$?

For $p=\infty$ such an $f$ exists by Arzela-Ascoli, but does this also hold for other $L_p$-norms?