= Solution
Write $E(t)=\int_{B_t(x_0)}|\nabla u|^2$. For $n\geq2$, choose $c$ in the <annular Caccioppoli inequality> to be the <integral average> of $u$ over $A_R$. The supplied <Poincare inequality on an annulus> gives
$$
E(R)\leq C_1R^{-2}\int_{A_R}|u-c|^2\leq C_2\int_{A_R}|\nabla u|^2=C_2\bigl(E(2R)-E(R)\bigr).
$$
Moving the $E(R)$ term to the left is the <hole-filling argument>:
$$
E(R)\leq\theta E(2R),\qquad\theta=\frac{C_2}{1+C_2}\in(0,1).
$$
Consequently $E(2^{-k}r_0)\leq\theta^kE(r_0)$. Put $\beta=-\log_2\theta>0$ and choose $\mu=\min\{\beta/2,1/2\}\in(0,1)$. For $2^{-k-1}r_0<r\leq2^{-k}r_0$, monotonicity gives
$$
E(r)\leq\theta^kE(r_0)\leq2^{-k\mu}E(r_0)\leq2^\mu(r/r_0)^\mu E(r_0).
$$
Dyadic endpoints can be assigned to either adjacent interval. Since $E(r_0)\leq\int_B|\nabla u|^2$, the requested <dyadic energy decay> is
$$
\boxed{E(r)\leq K(r/r_0)^\mu\int_B|\nabla u|^2,\qquad K=2^\mu.}
$$
Both constants depend only on the dimension and <uniform ellipticity> bounds, not on $u,x_0,r,r_0$.
There is a dimensional detail in the printed hint: an <annulus> is disconnected in dimension one, so that <Poincare inequality> with a single average is false there. The conclusion still holds. In one dimension the weak equation gives $a(x)u'(x)=J$ almost everywhere for a <constant flux for a one-dimensional divergence-form equation> $J$. Since $\lambda\leq a\leq\Lambda$,
$$
E(r)\leq\frac{2rJ^2}{\lambda^2},\qquad E(r_0)\geq\frac{2r_0J^2}{\Lambda^2},\qquad E(r)\leq(\Lambda/\lambda)^2(r/r_0)E(r_0).
$$
This implies the required estimate with, for example, $\mu=1/2$ and $K=(\Lambda/\lambda)^2$. If $J=0$, the estimate is immediate. Thus the proof also covers dimension one without using the inapplicable hint.
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