Let $1 < p < \infty$. Let $\{u_j\}_{j=1}^\infty$ be a sequence of functions in $W^{1,p}_{\text{loc}}(\mathbb{R}^n)$ such that
- $\nabla u_j \in L^p(\mathbb{R}^n)$ for all $j$,
- $\int_{B(0,1)} u_j dx = 0$ for all $j$,
- $\{ \nabla u_j \}_{j= 1}^\infty$ is a Cauchy sequence in $L^p(\mathbb{R}^n)$.
I would like to know whether $\{u_j\}_{j=1}^\infty$ is a Cauchy sequence in $L^p_{\text{loc}}(\mathbb{R}^n)$ (and hence convergent in $L^p_{\text{loc}}(\mathbb{R}^n)$ since this space is a Frechet space). That is, I would like to know whether it can be shown that, for any ball $B(0,r)$:
$$\|u_j - u_k \|_{L^p(B(0,r))}^p \to 0 $$
as $j$ and $k$ become large.
This result is claimed to be true in this paper on arXiv (see Proposition 2.1). The details of the proof are not supplied.
If 2. above is replaced by the condition that $\int_{B(0,r)} u_j dx = 0$ for all $r$ and all $j$, then I can get the result by applying Poincare's inequality for a ball (see Theorem 2 in section 5.8 from Evan's PDE book), and then and using 3. However, with only the weaker condition I'm not sure how to proceed.
Hints or solutions are greatly appreciated!
The vanishing integral over the small ball is enough to get a Poincaré-type estimate. Let $B = B(0,1)$ and $\Omega = B(0,r)$.
We define $$\|u\|_\star := \big|\int_B u \, \mathrm dx\big| + \|\nabla u\|_{L^p(\Omega)}.$$
It is clear that $\|\cdot\|_\star$ is a norm on $W^{1,p}(\Omega)$ and that $\|u\|_\star \le C \, \|u\|_{W^{1,p}(\Omega)}$ for some $C > 0$. Moreover, one can show that $W^{1,p}(\Omega)$ is complete w.r.t. $\|\cdot\|_\star$. From the bounded inverse theorem, we get a constant $M>0$ such that $\|u\|_{W^{1,p}(\Omega)} \le M \, \|u\|_\star$. Note that $M$ may (and, surely, will) depend on $r$, but this does not matter.