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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 $\xi \in {\mathbb{R}}^n$,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 isThen the Harnack inequality for uniformly elliptic divergence-form equations isA 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.

There is no need to assume that the original zeroth-order coefficient is nonpositive. Set $c_{+}=\max (c,0)$ and $\tilde{c}=\min (c,0)$, and use the uniformly elliptic operatorBecause $u\ge 0$ and $f\le 0$,Now $\tilde{c}\le 0$, so the strong minimum principle for elliptic operators applies. If $u$ vanished at an interior point of the connected domain, this principle would force $u$ to vanish identically. The given nontriviality therefore provesEvery nontrivial nonnegative solution is strictly positive inside. This uses the nondivergence-form strong minimum principle for elliptic operators; the divergence-form Harnack inequality for uniformly elliptic divergence-form equations in part (a) is not being applied to this operator.

Use the Weak Harnack inequality: for some $q>1$ and every nonnegative weak supersolution,where $C$ and $q$ depend only on $n,\lambda ,\Lambda$. A weak solution is both a subsolution and a supersolution. Applying part (i), after rescaling from $B_1$ to $B_{1/2}$, and then the weak Harnack inequality givesThis is the Harnack inequality for uniformly elliptic divergence-form equations.

Suppose $|Dv|\le M$. The coefficient matrices in part (ii) then have uniform ellipticity constants depending only on $M$. Applying the Harnack inequality for uniformly elliptic divergence-form equations to the nonnegative solutions $M+D_k{v}_R$ and $M-D_k{v}_R$ gives a scale-independent oscillation contractionScaling back,For fixed $r$, iterate this estimate with $R=4^{m}r$ and use the global bound $|D_{k}v|\le M$ to obtain ${\mathrm{osc}}_{B_r}{D}_{k}v\le 2M{\theta }^m\to 0$. Every partial derivative is therefore constant, so $v(x)=a\cdot x+b$ is an affine function. This is a bounded-gradient Bernstein theorem for entire minimal graphs.

The weak estimate for supersolutions is distinct from the sup-to-inf Harnack inequality for uniformly elliptic divergence-form equations for solutions.

For a nonnegative weak supersolution of a uniformly elliptic divergence-form equation, a positive $L^q$ mean on a ball is bounded by a constant times the infimum on a smaller concentric ball. The admissible exponent and constant depend only on dimension and the ellipticity bounds.