L^p bounds for function on double annulus

Question

The problem is
Consider the open disc $B_r=B(0,r)\subset \mathbb{R}^2$, $\phi=\max\{|x|-1,0\}$ is the distance from $B_1$.
For $u\in C^1(\overline{B_3-B_1})$. Show that $\exists$ $c$ such that
(i)$1\leq p<+\infty$, \begin{equation} \lVert u\rVert_{L^p(B_3-\overline{B_1})}\leq c\left(\lVert \phi\nabla u\rVert_{L^p(B_3-\overline{B_1})}+\lVert u\rVert_{L^p(B_3-\overline{B_2})}\right) \end{equation} (ii)$1\leq p<2$, \begin{equation} \lVert u\rVert_{L^{p^{*}}(B_3-\overline{B_1})}\leq c\left(\lVert \phi\nabla u\rVert_{L^p(B_3-\overline{B_1})}+\lVert u\rVert_{L^p(B_3-\overline{B_2})}\right) \end{equation} Here $p^{*}$ is dual of $p$ satisfying $\dfrac{1}{p}+\dfrac{1}{p^{*}}=1$.
Maybe the result can be shown for general compact sets $K\subset L$.