= Solution
Adopt the sign convention that a <weak supersolution> for $Lu=D_i(a^{ij}D_ju)$ satisfies $Lu\leq0$, or $\int a^{ij}D_juD_i\varphi\geq0$ for every nonnegative compactly supported <test function>. For a nonnegative supersolution, the <Weak Harnack inequality> gives some $q>0$ and $C_H\geq1$, depending only on $n,\lambda,\Lambda$, such that
$$
\boxed{\left(\frac1{|B_{1/2}|}\int_{B_{1/2}}u^q\,dx\right)^{1/q}
\leq C_H\operatorname*{ess\,inf}_{B_{1/2}}u.}
$$
Here $|E|^{-1}\int_E$ denotes the average over $E$; equivalently the right side gives a lower bound for the <essential infimum>. One may take a sufficiently small positive exponent, so no assumption $q\geq1$ is needed. The inequality is unchanged under translations and dilations of the <Euclidean balls>.
Since a <Sobolev space> element is defined up to changes on a null set, the infimum is initially essential. After choosing a continuous representative it is the ordinary infimum. Without choosing such a representative, an arbitrary pointwise infimum cannot be bounded by an integral.
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