= Solution
Assume $u\in W^{1,2}(B_2(0))$ is nonnegative <almost everywhere>, and that the real coefficient <matrix> $A=(a^{ij})$ is measurable and satisfies, for almost every $x$ and every $\xi\in\mathbb R^n$,
$$
\xi\cdot A(x)\xi\geq\lambda|\xi|^2,\qquad
|A(x)\xi|\leq\Lambda|\xi|,
\qquad 0<\lambda\leq\Lambda<\infty.
$$
The <operator norm> bound controls all coefficients, including any antisymmetric part. For a <symmetric matrix>, these assumptions are exactly the usual lower and upper quadratic-form bounds. The <weak solution> condition is
$$
\int_{B_2}a^{ij}D_j uD_i\zeta=0\qquad(\zeta\in C_c^\infty(B_2)).
$$
Then the <Harnack inequality for uniformly elliptic divergence-form equations> is
$$
\boxed{\operatorname*{ess\,sup}_{B_1}u\leq C(n,\Lambda/\lambda)\operatorname*{ess\,inf}_{B_1}u.}
$$
\b[A nonnegative weak solution has its supremum controlled by its infimum on the smaller ball.] The <De Giorgi-Nash-Moser theorem> supplies a <Hölder continuous function> representative, for which ordinary supremum and infimum can replace the <essential supremum> and <essential infimum>. Nonnegativity and the larger ball are essential hypotheses; there is no differentiability requirement on the coefficients.
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