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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.

A Weak Harnack inequality controls a positive integral mean of a nonnegative weak supersolution by its essential infimum. For example, let $n\ge 3$, let $A$ be a bounded measurable uniformly elliptic coefficient matrix, and let $u\in H_{\mathrm{loc}}^1$ be nonnegative with $\mathrm{div}(A\nabla u)\le 0$ in the weak sense. If $B_{4R}⋐\Omega$, then for $0<q<n/(n-2)$,The constant depends on dimension, ellipticity/coefficient bounds and the chosen exponent. The normalization makes the estimate invariant under spatial rescaling. Its infimum is essential because a function in a Sobolev space is initially an almost-everywhere equivalence class; changing a single representative value cannot invalidate the estimate.

This is a one-sided statement for supersolutions, distinct from the sup-to-inf Harnack inequality for uniformly elliptic divergence-form equations for solutions. Combining it with the subsolution bound from Moser iteration yields that latter estimate. If the essential infimum is zero, the left side forces vanishing almost everywhere on the larger ball, giving the mechanism of propagation of zeros by the weak Harnack inequality. Lower-order terms or forcing require the corresponding modified assumptions and error terms.