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If $Lu=\mathrm{div}F+g$ and $L$ contains $\mathrm{div}(bu)+du$, then $L(u-k)=\mathrm{div}(F-kb)+(g-kd)$. Thus shifting a weak solution to make it nonnegative changes the forcing unless the operator annihilates constants. This bookkeeping is essential in applying the Weak Harnack inequality to upper and lower oscillations.

A nonnegative weak solution of a uniformly elliptic divergence-form equation satisfieswhenever a fixed larger concentric ball lies in the domain. Combine the subsolution bound from Moser iteration with the Weak Harnack inequality for supersolutions.

For a weak solution of a uniformly elliptic divergence-form equation, local boundedness and the Weak Harnack inequality implyfor a universal $0<\theta <1$. Iteration gives local Hölder continuity.

Put $\alpha =1-n/q>0$ andFirst suppose $A<\infty$. For an interior ball $B_R$, with $R\le 1$, write $M_R={\mathrm{esssup}}_{B_R}u$, $m_R={\mathrm{essinf}}_{B_R}u$, and $\omega (R)=M_R-m_R$. Constant shifts of a divergence-form elliptic operator change its forcing. The two nonnegative weak supersolutions $v=M_R-u$ and $w=u-m_R$ satisfy, respectively,Their Weak Harnack inequalities on $B_{R/2}$ therefore have forcing errors bounded by $CN{R}^{\alpha }$. Indeed $|M_R|,|m_R|\le A$, the new divergence forcing is bounded by $\Vert F{\Vert }_q+A\Vert b{\Vert }_q$, and the scalar forcing by $\Vert g{\Vert }_{q/2}+A\Vert d{\Vert }_{q/2}$; also $R^{2\alpha }\le R^{\alpha }$.

On $B_{R/2}$ we have $v+w=\omega (R)$. Even if $p<1$, the elementary inequality $(s+t{)}^p\le C_p(s^p+t^p)$ impliesThe sum of the two essential infima is $\omega (R)-\omega (R/2)$. Enlarging a uniform constant $C_0>1$ giveswhere $\theta =1-C_0^{-1}\in (0,1)$. This is an oscillation decay estimate; inhomogeneous oscillation decay implies Hölder continuity. ChooseIterating on concentric balls, starting at a fixed admissible radius $R_{\ast }\le 1$, yieldsThe strict choice of $\mu$ bounds the geometric series, including the case in which the two original decay rates coincide. Monotonicity of oscillation extends this bound to intermediate radii.

Take $R_{\ast }$ uniformly smaller than the distance of ${\Omega }^{\prime }$ from the boundary of $\Omega$. The vanishing essential oscillation produces a continuous representative of $u$. A ball centred at either of two sufficiently close points of ${\Omega }^{\prime }$ contains both, so its oscillation bounds their difference by $CN|x-y{|}^{\mu }$. For separated points use $|u(x)-u(y)|\le 2A$. Thus the Hölder seminorm satisfiesThe constants depend only on the indicated domain separation, dimension, coefficient norms, uniform ellipticity bounds and $q$. If the global essential supremum is infinite, the displayed global estimate is vacuous; local boundedness of weak elliptic subsolutions on slightly larger compact subsets still makes the same argument prove interior Hölder continuity.

Let $M={\mathrm{esssup}}_{\Omega }u$. The standard local boundedness of weak elliptic subsolutions gives a finite upper bound on every relatively compact ball. In particular, the assumed equality of essential suprema makes $M$ finite. Equivalently, when using the maximum principle for an upper-bounded weak subsolution, this preliminary fact is already part of the hypothesis.

Set $v=M-u$. Because the divergence-form elliptic operator here annihilates constants, $v\ge 0$ and $Lv=-Lu\le 0$. The hypothesis says ${\mathrm{essinf}}_{B}v=0$. Cover the compact closure of $B$ by finitely many balls $D_j$ whose doubled balls lie in $\Omega$. At least one $D_j$ has essential infimum zero: otherwise the minimum of their finitely many positive lower bounds would be a positive lower bound on $B$. The Weak Harnack inequality with zero forcing now givesConsequently $v=0$ almost everywhere on $D_j$. If another admissible ball overlaps $D_j$ in an open set, its essential infimum of $v$ is also zero, and the same argument makes $v$ vanish there. Since a domain is connected, a chain of overlapping interior balls reaches every point of $\Omega$. This propagation of zeros by the weak Harnack inequality proves $u=M$ almost everywhere on $\Omega$, which is the claimed strong maximum principle. Notice why the absence of the $b,d$ terms matters: it is precisely what makes $L(M-u)=-Lu$.

The Weak Harnack inequality is an estimate for a nonnegative weak supersolution. Write $F=(f^1,\dots ,f^n)$, let $B_{2R}(x_0)\subset \subset \Omega$, and assume $u\ge 0$ almost everywhere on this ball. There are $p>0$ and $C<\infty$, depending only on the dimension, the uniform ellipticity bounds, $q$, and the scaled norms of the lower-order coefficients, such thatHere $⟨v{⟩}_E=|E{|}^{-1}{\int }_{E}v$ is the integral average. The weak supersolution inequality uses the sign convention $Lu\le \mathrm{div}F+g$. For instance, the coefficient dependence can be expressed usingThus one may use the same $p,C$ on all sufficiently small balls when the global coefficient norms and uniform ellipticity bounds are fixed. The nonnegativity condition is essential to this formulation; a signed weak supersolution may first be shifted, with the resulting change in its forcing included. As usual, supremum and infimum statements for functions in a Sobolev space mean essential suprema and essential infima. The standard multidimensional statement uses $n\ge 2$, so $q/2>1$. In dimension one the analogous formulation requires $q\ge 2$; the printed restriction $q>n$ alone would allow $q/2<1$, which does not suffice. To see the obstruction, smooth the nonnegative capped function $\min \{1,|x|/\epsilon \}$ by a mollifier of width $\tau$. Its positive second derivative is a bump of mass $2/\epsilon$ and has $L^{q/2}$ size of order ${\epsilon }^{-1}{\tau }^{2/q-1}$. For fixed $\epsilon$ and $1<q<2$, this size and the minimum both tend to zero as $\tau \downarrow 0$, whereas the average on a fixed larger interval stays positive. Thus the Weak Harnack inequality cannot have the displayed uniform forcing bound in that range. Shrinking $\epsilon$ and choosing $\tau$ still smaller likewise defeats a uniform Hölder estimate based on that norm alone.

Adopt the sign convention that a weak supersolution for $Lu=D_i(a^{ij}{D}_{j}u)$ satisfies $Lu\le 0$, or $\int a^{ij}{D}_{j}u{D}_i\phi \ge 0$ for every nonnegative compactly supported test function. For a nonnegative supersolution, the Weak Harnack inequality gives some $q>0$ and $C_H\ge 1$, depending only on $n,\lambda ,\Lambda$, such thatHere $|E{|}^{-1}{\int }_E$ 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 $q\ge 1$ 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.

Work first with essential bounds. On a Euclidean ball $B_R(x_0)⋐B_1$, put $M={\mathrm{esssup}}_{B_R}u$, $m={\mathrm{essinf}}_{B_R}u$ and $\omega =M-m$. The functions $M-u$ and $u-m$ are nonnegative weak solutions, hence weak supersolutions. In $B_{R/2}$, at least one of the sets $\{u\le (M+m)/2\}$ and $\{u\ge (M+m)/2\}$ has at least half the measure.

In the first case, the Weak Harnack inequality applied to $M-u$ givesIn the second case, applying it to $u-m$ gives the same lower bound for the improvement of the minimum. Thus the oscillation decay estimate isThis measure argument works even when $q<1$, avoiding a false use of the triangle inequality in that range.

Iteration yields ${\mathrm{osc}}_{B_r(x_0)}u\le C(r/R{)}^{\mu }{\mathrm{osc}}_{B_R(x_0)}u$, with, for example, $\mu =\min \{1/2,-\log \tau /\log 2\}\in (0,1)$. At Lebesgue points the shrinking oscillations define a unique continuous representative, and the estimate extends to every point by continuity. For $x,y\in B_{1/4}$ at distance less than $1/4$, apply the estimate to a Euclidean ball centred at $x$ with initial radius $1/2$ and final radius comparable to $|x-y|$. This Euclidean ball stays in $B_1$. Larger distances use the trivial bound $2\Vert u{\Vert }_{\infty }$. We obtainThe constants depend only on $n,\lambda ,\Lambda$. This derives the relevant conclusion of the De Giorgi-Nash-Moser theorem directly from part (a).

The Weak Harnack inequality states that there are $p>0$ and $C=C(n,\lambda ,\Lambda )$ such that every nonnegative weak supersolution satisfiesThe radii may be replaced by any fixed nested pair of balls, with the constant adjusted accordingly.

On a ball $B_{2r}(x_0)⋐B$, bothare nonnegative solutions and hence supersolutions. Combining the Weak Harnack inequality with the local boundedness estimate for subsolutions gives an oscillation decay estimatewhere $\theta$ depends only on $n,\lambda ,\Lambda$. Iteration yields Hölder continuity with some exponent $\mu \in (0,1)$. The local $L^2$-to-$L^{\infty }$ estimate controls the initial oscillation and gives

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.

For a nonnegative weak supersolution with zero forcing, the Weak Harnack inequality makes zero essential infimum on an interior ball imply vanishing almost everywhere on that ball. An overlapping ball then has zero essential infimum too. Chains of overlapping interior balls propagate the vanishing through a connected domain.

For $Lu=D_i(a^{ij}{D}_{j}u)$, a weak supersolution satisfies $Lu\le 0$ as a distribution, equivalently $\int a^{ij}{D}_{j}u{D}_i\varphi \ge 0$ for every nonnegative compactly supported test function $\varphi$. The Weak Harnack inequality applies to nonnegative weak supersolutions with this sign convention.