Why do we deduce ${\vert \vert f_n \vert \vert}_{W^{1,p}(\Gamma)} \to 0$?

Question

I'm having a really hard time understanding the following implication in the paper I 'm at the moment reading.

Since $f_n \to 0$ in $L^1(\Gamma)$ and $f_n$ is uniformly bounded in $W^{2,p}(\Gamma)$, then ${\vert \vert f_n \vert \vert}_{W^{1,p}(\Gamma)} \to 0$ which follows from embedding and interpolation

($\Gamma$ here denotes the $C^2-$regular boundary of a bounded,open and connected subset of $\mathbb R^3$)

So by Kondrachov embedding theorem we have that $f_n$ is also uniformly bounded in $W^{1,p}(\Gamma)$. Then maybe I could use the Rellich's theorem but I don't see the point in it since all I would get is the $L^p-$norm bounded by the $W^{1,p}$ while I suppose I need the inverse. In addition I 'm not familiar to the interpolation in Sobolev spaces and probably this has a key role here.

I would really appreciate if somebody could help me through this. What Theorems do I miss?

Thanks in advance!

Answer 1

Part 1: Let us first show the result for $\Gamma = \mathbb R^2$: Let $f_n \in W^{2,p}(\Gamma)$ be such that $\|f_n\|_{W^{2,p}(\Gamma)}$ is uniformly bounded and $f_n \to 0$ in $L^1(\Gamma)$.

By the Gagliardo–Nirenberg inequality, see E.q. 1.1. in this paper we have there exists C>0 and $0<\theta<1$ such that $$ \| f_n \|_{W^{1,p}(\Gamma)} \leq C \| f_n\|_{W^{2,p}(\Gamma)}^\theta \| f_n\|_{L^1(\Gamma)}^{1-\theta}. $$ Hence, by the uniform boundedness of $\| f_n\|_{W^{2,p}(\Gamma)}^\theta$ and the fact that $f_n \to 0 $ in $L^1$ we conclude that $f_n \to 0 $ in $W^{1,p}$.

Using, for example Proposition 2.3 of the paper cited above, yields the same result for $\Gamma = (0,1)^2$

Part 2: Next we assume that $\Gamma$ is a compact $C^2$ regular subset of $\mathbb R^3$, then this means that for every point $x \in \Gamma$ there is a neighborhoud $U_x\subset \Gamma$ and a $C^2$ chart $\gamma_x: (0,1)^2 \to U_x$ such that $f_n \circ \gamma_x \in W^{2,p}((0,1)^2)$ where $\|f_n \circ \gamma_x\|_{W^{2,p}}$ is bounded independent from $n$ and $\|f_n \circ \gamma_x\|_{L^1} \to 0$. By the first part of the proof, we have that $$ \|f_n \circ \gamma_x\|_{W^{1,p}((0,1)^2)} \to 0, $$ and therefore $$ \|(f_n)_{|U_x}\|_{W^{1,p}(U_x)} \to 0, $$ for all $x \in \Gamma$. Moreover, $\|f_n\|_{W^{1,p}(\Gamma)} \to 0$, by a compactness argument.