02.17.08 · analysis / elliptic-regularity

The Harnack Inequality for Elliptic Equations (Moser and Krylov-Safonov)

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Anchor (Master): Gilbarg-Trudinger §8.6-§8.10, §9.7-§9.8; Moser, On Harnack's theorem for elliptic differential equations (CPAM 14, 1961); Krylov-Safonov, Certain properties of solutions of parabolic equations with measurable coefficients (Izv. 1980); Caffarelli-Cabré §4; Han-Lin §4

Intuition Beginner

Picture a steady temperature inside a block of material, with no heat being added or removed in the interior, and suppose the temperature never drops below zero anywhere. The Harnack inequality makes a striking promise about such a steady state: across any region sitting comfortably inside the block, the hottest spot cannot be more than a fixed multiple of the coldest spot. If somewhere the temperature is warm, then everywhere nearby it must be at least lukewarm; a positive equilibrium cannot pair a towering peak with a deep valley right beside it. The peak and the valley are tied together by one universal number.

Why should this be true? A steady state with no interior sources is a balancing act: the value at each point is forced to be a weighted average of the values around it. Averaging refuses to leave a hole next to a hill. If the temperature were tiny at one interior point, the averaging would drag down all its neighbours, and theirs in turn, so the small value would spread out and cap the whole region from above. A high value somewhere therefore guarantees the low value cannot be too low. The single multiplier that quantifies "not too low" is the Harnack constant.

The remarkable part is how little the material needs to cooperate. The block can be a chaotic patchwork whose conductivity jumps wildly from point to point, with no smoothness and no pattern at all. The Harnack constant still exists, and it depends only on how far the conductivity is allowed to swing between its fixed upper and lower limits, never on the actual mess of its pattern.

The one-sentence takeaway: for a non-negative interior equilibrium, the largest value over a region is bounded by a fixed multiple of the smallest value over that region, and the multiplier depends only on the dimension and the conductivity bounds, not on the chaotic details of the medium.

Visual Beginner

The picture to hold is a positive landscape pinned between two horizontal ceilings whose heights are locked at a fixed ratio by the value at a single point.

The two ceilings are the whole statement. The lower ceiling sits at the smallest value the solution takes on the region; the upper ceiling sits at the largest. Harnack says the gap between them is controlled: the upper ceiling is at most a fixed multiple of the lower one. The crossed-out side panel is the shape Harnack forbids, a high peak next to an almost-zero dip, because that would mean the largest value is a huge multiple of the smallest. The checkerboard panel reminds you that the medium underneath can be as rough as you please; the trapping between ceilings happens anyway.

Worked example Beginner

We turn the Harnack multiplier into a concrete pair of bounds, and then watch how it forbids a peak beside a valley. Suppose we have a non-negative steady state on a region, and we have been told its Harnack constant is $4$ : the largest value over the region is at most $4$ times the smallest value over the region.

Step 1. Pin one value. Suppose we measure the temperature at one interior point and find it is $10$ degrees. The smallest value over the whole region is therefore at most $10$ , since that one point is part of the region.

Step 2. Bound the peak from the valley. Suppose instead we are told the smallest value over the region is exactly $3$ degrees. Harnack says the largest value is at most $4$ times $3$ , which is $12$ degrees. So the whole region is trapped between $3$ and $12$ .

Step 3. Bound the valley from the peak. Now suppose we are told the largest value is $20$ degrees. Harnack rearranges to say the smallest value is at least the largest divided by $4$ , which is $20$ divided by $4$ , or $5$ degrees. A warm peak forces a warm floor.

Step 4. See the forbidden shape. Could the temperature be $20$ at one point and $1$ at another point in the same region? That would need the largest value to be at least $20$ times the smallest, which is $20$ . But the Harnack constant is $4$ , and $20$ is bigger than $4$ . So this peak-beside-valley shape is impossible.

Step 5. Read the rate of spread. The same multiplier $4$ holds on every region of the right kind, no matter where we centre it or how small we make it. That uniformity is what later forces the solution to be continuous: shrinking the region squeezes the gap between the ceilings.

What this tells us: a single multiplier turns one measured value into a two-sided bound on the whole region, and it rules out any arrangement where a high value and a near-zero value sit close together. The multiplier is fixed in advance by the dimension and the conductivity bounds, so the same number does this job everywhere.

Check your understanding Beginner

Exercise (easy, multiple choice).

The Harnack inequality for a non-negative interior equilibrium says that over a region sitting inside the domain:

A. the solution is constant B. the largest value is at most a fixed multiple of the smallest value C. the solution equals its boundary values D. the smallest value is zero

Hint

It ties the peak and the valley together by one universal number.

Answer

B. the largest value is at most a fixed multiple of the smallest value. Harnack pins the sup to the inf by a single constant that depends only on the dimension and the conductivity bounds. Feedback-correct: exactly — a warm spot forces a warm floor nearby, and a high peak cannot sit beside a near-zero valley. Feedback-wrong: A is far too strong (the solution can vary, just not too violently); C confuses an interior estimate with the boundary data; D is false, since a positive solution stays strictly positive in the interior.

Formal definition Intermediate+

Two distinct operator classes are in play, and the Harnack inequality holds for both, by genuinely different proofs.

For the divergence-form (Moser) theory, let $Ω \subseteq R^{n}$ be open, $n \geq 2$ , and let $A : Ω \to R^{n \times n}$ have bounded measurable entries $a^{ij}$ , uniformly elliptic: $λ ∣ ξ ∣^{2} \leq i, j \sum a^{ij} (x) ξ_{i} ξ_{j} \leq Λ∣ ξ ∣^{2} for a.e. x, all ξ \in R^{n}, 0 < λ \leq Λ.$ The operator $Lu = - div (A D u)$ is read weakly: $u \in W^{1, 2} (Ω)$ 24.01.01 is a weak solution of $Lu = 0$ when $\int_{Ω} \sum_{i, j} a^{ij} D_{j} u D_{i} φ d x = 0$ for all $φ \in C_{c}^{\infty} (Ω)$ , with the subsolution ( $\leq 0$ ) and supersolution ( $\geq 0$ ) variants tested against $φ \geq 0$ ^{[Gilbarg-Trudinger 2001 §8.6]} 02.17.07.

For the non-divergence (Krylov-Safonov) theory, the operator is $Lu = i, j \sum a^{ij} (x) D_{ij} u, λ ∣ ξ ∣^{2} \leq a^{ij} ξ_{i} ξ_{j} \leq Λ∣ ξ ∣^{2},$ with $a^{ij}$ symmetric, bounded, measurable 02.17.03. Here $Lu$ cannot be integrated by parts, so the relevant notion is the strong solution ( $u \in W_{loc}^{2, n}$ , equation holding a.e.) or the $L^{n}$ -viscosity solution; the Pucci extremal operators $M_{λ, Λ}^{-} (M) = λ e_{i} > 0 \sum e_{i} + Λ e_{i} < 0 \sum e_{i}, M_{λ, Λ}^{+} (M) = Λ e_{i} > 0 \sum e_{i} + λ e_{i} < 0 \sum e_{i}$ (with $e_{i}$ the eigenvalues of the symmetric matrix $M$ ) sandwich every operator in the class: $M^{-} (D^{2} u) \leq Lu \leq M^{+} (D^{2} u)$ .

Definition (Harnack inequality). A class of non-negative solutions on $B_{4 R} (x_{0}) ⋐ Ω$ satisfies the Harnack inequality when there is a constant $C = C (n, Λ/ λ)$ , independent of $u$ , $x_{0}$ , and $R$ , with $B_{R} (x_{0}) sup u \leq C B_{R} (x_{0}) in f u .$ The constant $C$ is scale-invariant (unchanged under $u (x) \mapsto u (x_{0} + R x)$ ) and sees the coefficients only through the ellipticity ratio $Λ/ λ$ . The inequality has a sup half (local boundedness, $sup u ≲$ an average of $u$ ) and an inf half (the weak Harnack inequality, an average of $u ≲ in f u$ ); their conjunction is the full statement.

The contrast with the harmonic case is the point. For $L = Δ$ the classical Harnack inequality of 02.13.01 follows from the Poisson kernel and the mean-value property, both of which are explicit. With merely measurable coefficients there is no kernel and no mean-value identity; the inequality must be extracted from the structure of sub- and super-solutions alone. The two routes do this differently: Moser iterates $L^{p}$ norms and crosses $p = 0$ through the logarithm; Krylov and Safonov run a measure-theoretic growth lemma off the ABP estimate.

Counterexamples to common slips Intermediate+

Non-negativity is essential. Harnack is false for sign-changing solutions: $u (x) = x_{1}$ is harmonic on any ball but $sup u / in f u$ is negative or undefined. The estimate compares a positive sup to a positive inf; without a sign the comparison is vacuous.
The inner ball must sit strictly inside. The inequality holds on $B_{R}$ with room to spare ( $B_{4 R} ⋐ Ω$ ); it fails up to the boundary, where a positive solution can vanish (consider $u$ harmonic and positive inside, zero at a boundary point). The constant degenerates as the inner ball approaches $\partial Ω$ .
The two operator forms need different machinery. $- div (A D u)$ admits integration by parts and Caccioppoli energy estimates; $a^{ij} D_{ij} u$ does not. Applying Moser iteration to the non-divergence operator is illegitimate because the weak formulation that powers every Moser step is unavailable; the Krylov-Safonov ABP route is required instead.
The constant blows up with the ellipticity ratio. $C = C (n, Λ/ λ)$ grows without bound as $Λ/ λ \to \infty$ . There is no Harnack inequality uniform over all elliptic operators; the bound on the ratio is what makes the class tractable.

Key theorem with proof Intermediate+

Theorem (Moser's weak Harnack inequality, cross-over form). Let $u \in W^{1, 2} (Ω)$ , $u \geq 0$ , be a weak supersolution of $- div (A D u) = 0$ with $A$ uniformly elliptic. There exist $p_{0} = p_{0} (n, Λ/ λ) > 0$ and $C = C (n, Λ/ λ)$ such that for every $B_{4 R} (x_{0}) ⋐ Ω$ , $(\fint_{B_{2 R}} u^{p_{0}})^{1/ p_{0}} \leq C B_{R} in f u$ ^{[Moser 1961]} ^{[Gilbarg-Trudinger 2001 §8.18]}. The crux is the passage from positive powers $u^{p}$ to negative powers $u^{- q}$ across the singular exponent $p = 0$ , supplied by the John-Nirenberg lemma applied to $lo g u$ .

Proof. Normalise $R = 1$ , $x_{0} = 0$ by scaling; the constants depend only on $n, Λ/ λ$ . Replace $u$ by $u + δ$ for $δ > 0$ to make it strictly positive, and remove $δ$ at the end. Three ingredients combine.

Step 1: Moser iteration on positive powers. For a supersolution $u > 0$ and any exponent $s < 1$ , $s \neq = 0$ , the function $u^{s}$ is a subsolution-type quantity after the chain rule, and testing the supersolution against $η^{2} u^{2 s - 1}$ together with the Sobolev inequality 02.16.01 yields a reverse-Hölder gain $(\fint_{B_{r^{'}}} u^{sχ})^{1/ (sχ)} \leq C (s) (\fint_{B_{r}} u^{s})^{1/ s}, χ = \frac{n}{n - 2} > 1,$ for $0 < r^{'} < r \leq 2$ , exactly as in De Giorgi-Nash-Moser 02.17.07. Iterating over $s_{k} = s χ^{k}$ for any fixed $0 < s < p_{0}$ propagates the bound from the exponent $s$ up to $+ \infty$ for the negative-power range and down for the positive range; the upshot is two one-sided estimates, $(\fint_{B_{3/2}} u^{p})^{1/ p} \leq C (\fint_{B_{2}} u^{p_{0}})^{1/ p_{0}} (0 < p \leq p_{0}), B_{1} in f u \geq c (\fint_{B_{3/2}} u^{- q})^{- 1/ q} (q > 0),$ the first controlling small positive powers from above, the second controlling $in f u$ from below by a negative-power average.

Step 2: the logarithm of a supersolution has bounded mean oscillation. Test the supersolution inequality $\int A D u \cdot D φ \geq 0$ against $φ = η^{2} u^{- 1}$ . With $w = lo g u$ and $D w = u^{- 1} D u$ , the computation of [02.17.07, Proposition 1] gives the logarithmic Caccioppoli estimate $\int_{B_{r}} ∣ D w ∣^{2} \leq C (n, Λ/ λ) r^{n - 2}$ . By the Poincaré inequality this bounds the mean oscillation of $w$ on every ball: $\fint_{B_{r}} ∣ w - (w)_{B_{r}} ∣ \leq C$ , so $w = lo g u \in BMO (B_{3/2})$ with a norm depending only on $n, Λ/ λ$ .

Step 3: John-Nirenberg crosses $p = 0$ . The John-Nirenberg inequality ^{[John-Nirenberg 1961]} states that a $BMO$ function has exponentially integrable oscillation: there are $p_{0}, C_{0} > 0$ depending only on its $BMO$ norm with $\fint_{B_{3/2}} e^{p_{0} ∣ w - (w)_{B_{3/2}} ∣} d x \leq C_{0} .$ Splitting the exponential into its two signs and writing $a = e^{(w)_{B_{3/2}}}$ , $\fint_{B_{3/2}} u^{p_{0}} \leq C_{0} a^{p_{0}} and \fint_{B_{3/2}} u^{- p_{0}} \leq C_{0} a^{- p_{0}},$ so $(\fint u^{p_{0}})^{1/ p_{0}} (\fint u^{- p_{0}})^{1/ p_{0}} \leq C_{0}^{2/ p_{0}}$ . This is the bridge: it links a positive-power average to a negative-power average through the common centre $a$ , precisely the link that no purely positive-power iteration can make.

Step 4: assemble. Chain the three steps. By Step 1 (positive range) and Step 3, $(\fint_{B_{3/2}} u^{p_{0}})^{1/ p_{0}} \leq C_{0}^{2/ p_{0}} (\fint_{B_{3/2}} u^{- p_{0}})^{- 1/ p_{0}} \leq C_{0}^{2/ p_{0}} C^{- 1} B_{1} in f u^{- 1} \cdot (Step 1, negative range),$ and tracking the constants gives $(\fint_{B_{3/2}} u^{p_{0}})^{1/ p_{0}} \leq C in f_{B_{1}} u$ . Letting $δ \to 0$ removes the regularisation, proving the weak Harnack inequality with $C, p_{0}$ depending only on $n, Λ/ λ$ . $□$

Bridge. This is the inf half of Harnack, and it is the foundational reason the divergence-form theory closes: Moser iteration alone produces only same-sign power bounds, and the John-Nirenberg lemma is exactly the device that ferries control across the singular exponent $p = 0$ , joining the positive-power average to the negative-power average that pins $in f u$ . This builds toward the full Harnack inequality, obtained by pairing this inf half with the sup half (local boundedness, 02.17.07) at the shared exponent $p_{0}$ , and it appears again in the oscillation-decay deduction below, where the weak Harnack inequality applied to $M - u$ and $u - m$ forces the spread of a solution to contract. The central insight is that the logarithm converts the multiplicative Harnack comparison into an additive $BMO$ bound, which is dual to the way the ABP estimate of 02.17.03 converts a pointwise touching into a measure estimate; this is exactly the divergence-form counterpart of the Krylov-Safonov growth lemma. Putting these together, the harmonic Harnack inequality of 02.13.01, proved there from the explicit Poisson kernel, survives the total loss of coefficient regularity because the kernel is replaced by the self-improving energy-plus- $BMO$ mechanism, and the bridge is the shared exponent $p_{0}$ at which the sup and inf halves meet.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Show that the John-Nirenberg cross-over estimate $(\fint_{B} u^{p_{0}})^{1/ p_{0}} (\fint_{B} u^{- p_{0}})^{1/ p_{0}} \leq C_{0}^{2/ p_{0}}$ follows from $\fint_{B} e^{p_{0} ∣ w - (w)_{B} ∣} \leq C_{0}$ with $w = lo g u$ .

Hint

$u^{\pm p_{0}} = e^{\pm p_{0} w} = a^{\pm p_{0}} e^{\pm p_{0} (w - (w)_{B})}$ where $a = e^{(w)_{B}}$ , and $e^{\pm p_{0} t} \leq e^{p_{0} ∣ t ∣}$ .

Answer

Write $a = e^{(w)_{B}}$ and $t = w - (w)_{B}$ , so $u^{p_{0}} = e^{p_{0} w} = a^{p_{0}} e^{p_{0} t}$ and $u^{- p_{0}} = a^{- p_{0}} e^{- p_{0} t}$ . Since $e^{\pm p_{0} t} \leq e^{p_{0} ∣ t ∣}$ pointwise, $\fint_{B} u^{p_{0}} = a^{p_{0}} \fint_{B} e^{p_{0} t} \leq a^{p_{0}} \fint_{B} e^{p_{0} ∣ t ∣} \leq a^{p_{0}} C_{0},$ and identically $\fint_{B} u^{- p_{0}} \leq a^{- p_{0}} C_{0}$ . Multiplying, the factors $a^{p_{0}}$ and $a^{- p_{0}}$ cancel: $(\fint_{B} u^{p_{0}}) (\fint_{B} u^{- p_{0}}) \leq C_{0}^{2}$ . Taking the $1/ p_{0}$ power gives the stated bound. The cancellation of $a$ is the whole mechanism: the unknown mean of $lo g u$ drops out, leaving a pure cross-over constant.

Exercise 4 (medium, symbolic).

Derive the full Harnack inequality $sup_{B_{R}} u \leq C in f_{B_{R}} u$ for non-negative weak solutions from the two halves: local boundedness $sup_{B_{R}} u \leq C_{1} (\fint_{B_{2 R}} u^{p_{0}})^{1/ p_{0}}$ (sup half) and the weak Harnack inequality $(\fint_{B_{2 R}} u^{p_{0}})^{1/ p_{0}} \leq C_{2} in f_{B_{R}} u$ (inf half).

Hint

A solution is both a subsolution and a supersolution; chain the two inequalities at the common exponent $p_{0}$ .

Answer

A weak solution is simultaneously a subsolution (so the sup-half local boundedness applies) and a supersolution (so the inf-half weak Harnack applies), and both are stated at the same exponent $p_{0}$ . Chain them: $B_{R} sup u \leq C_{1} (\fint_{B_{2 R}} u^{p_{0}})^{1/ p_{0}} \leq C_{1} C_{2} B_{R} in f u .$ Setting $C = C_{1} C_{2} = C (n, Λ/ λ)$ gives the Harnack inequality. The shared exponent $p_{0}$ is essential: it is the value at which the John-Nirenberg cross-over fixes the integrability, so the two halves meet without a gap.

Exercise 5 (medium, symbolic).

Deduce oscillation decay from Harnack. Let $u$ solve $Lu = 0$ on $B_{4 R} (x_{0})$ with $M = sup_{B_{4 R}} u$ , $m = in f_{B_{4 R}} u$ , and suppose the Harnack inequality $sup_{B_{R}} v \leq C in f_{B_{R}} v$ holds for non-negative solutions. Show $osc_{B_{R}} u \leq θ osc_{B_{4 R}} u$ with $θ = (C - 1) / (C + 1) < 1$ .

Hint

Apply Harnack to the two non-negative solutions $M - u$ and $u - m$ on $B_{R}$ , then add.

Answer

Both $M - u \geq 0$ and $u - m \geq 0$ solve $Lv = 0$ on $B_{4 R}$ (constants are annihilated by $div (A D \cdot)$ ). Harnack on $B_{R}$ gives $M - B_{R} in f u = B_{R} sup (M - u) \leq C B_{R} in f (M - u) = C (M - B_{R} sup u),$ $B_{R} sup u - m = B_{R} sup (u - m) \leq C B_{R} in f (u - m) = C (B_{R} in f u - m) .$ Add: the left side is $(M - m) + osc_{B_{R}} u$ , the right is $C [(M - m) - osc_{B_{R}} u]$ . Hence $osc_{B_{4 R}} u + osc_{B_{R}} u \leq C osc_{B_{4 R}} u - C osc_{B_{R}} u$ , which rearranges to $(C + 1) osc_{B_{R}} u \leq (C - 1) osc_{B_{4 R}} u$ , i.e. $θ = (C - 1) / (C + 1) < 1$ .

Exercise 7 (hard, symbolic).

State and sketch the Krylov-Safonov growth (measure) lemma and explain how it yields the weak Harnack inequality for the non-divergence operator. Specifically: let $u \geq 0$ be a supersolution ( $M^{-} (D^{2} u) \leq 0$ in the viscosity sense) on $B_{2 n}$ with $in f_{B_{1/4}} u \leq 1$ . Explain why there are $ε, C, M > 0$ depending only on $n, Λ/ λ$ with $∣ {x \in B_{1} : u > t} ∣ \leq C t^{- ε}$ for $t > 0$ , and how this decay gives $(\fint_{B_{1}} u^{ε})^{1/ ε} \leq C^{'} in f_{B_{1}} u$ .

Hint

The ABP estimate 02.17.03, applied on each cube of a Calderón-Zygmund decomposition where a paraboloid touches $u$ from below, converts pointwise touching into a fixed-proportion measure bound; iterating the decomposition compounds the proportion into power-law decay, which is the weak- $L^{ε}$ form of the average bound.

Answer

The growth lemma is the statement that if a non-negative supersolution is somewhere small ( $in f_{B_{1/4}} u \leq 1$ ) then it cannot be large on more than a controlled fraction of $B_{1}$ : the super-level sets ${u > t}$ have measure decaying like $t^{- ε}$ . It is proved by sliding paraboloids of fixed opening $M$ from below under $u$ over a Calderón-Zygmund dyadic cube decomposition. On each cube where a paraboloid touches, the ABP estimate 02.17.03 applied to $- u$ (whose extremal inequality $M^{-} (D^{2} u) \leq 0$ controls the contact-set Hessian) bounds the measure of the touching region below by a fixed proportion of the cube; the untouched cubes are subdivided and the argument iterates. Each iteration multiplies the measure of ${u > M^{k}}$ by a factor $μ < 1$ , so $∣ {u > M^{k}} \cap B_{1} ∣ \leq μ^{k}$ , which is exactly $∣ {u > t} \cap B_{1} ∣ \leq C t^{- ε}$ with $ε = lo g (1/ μ) / lo g M$ .

To pass from this weak- $L^{ε}$ decay to the average bound, integrate by the layer-cake formula: $\fint_{B_{1}} u^{ε /2} = \frac{ε}{2} \int_{0}^{\infty} t^{ε /2 - 1} ∣ {u > t} \cap B_{1} ∣ \frac{d t}{∣ B _{1} ∣} \leq C (1 + \int_{1}^{\infty} t^{ε /2 - 1} t^{- ε} d t) < \infty,$ the integral converging because $ε /2 - 1 - ε < - 1$ . Normalising out $in f_{B_{1}} u$ (the hypothesis $in f_{B_{1/4}} u \leq 1$ is the scale-fixing that makes the bound homogeneous of degree one) gives $(\fint_{B_{1}} u^{ε /2})^{2/ ε} \leq C^{'} in f_{B_{1}} u$ , the weak Harnack inequality with exponent $p_{0} = ε /2$ . The full Harnack inequality then follows by pairing with the local maximum principle for subsolutions, proved by the same paraboloid/ABP measure machinery applied to $M^{+}$ .

Exercise 8 (hard, symbolic).

Prove the Liouville theorem from the Harnack inequality: if $u$ is a weak solution of $- div (A D u) = 0$ on all of $R^{n}$ ( $A$ uniformly elliptic, bounded measurable) and $u$ is bounded below (say $u \geq 0$ ) and bounded above, then $u$ is constant.

Hint

Apply Harnack on $B_{R} (0)$ for $R \to \infty$ to $u - in f_{R^{n}} u$ and let the radius grow; the scale-invariant constant does not change.

Answer

Let $m = in f_{R^{n}} u$ (finite, since $u$ is bounded below) and set $v = u - m \geq 0$ , a non-negative solution on all of $R^{n}$ (constants are annihilated by the divergence operator). Fix $R > 0$ . The Harnack inequality on $B_{R} (0)$ , with its scale-invariant constant $C = C (n, Λ/ λ)$ independent of $R$ , gives $sup_{B_{R}} v \leq C in f_{B_{R}} v$ . Now let $R \to \infty$ : $sup_{B_{R}} v \to sup_{R^{n}} v$ and $in f_{B_{R}} v \to in f_{R^{n}} v = 0$ (because $m$ is the infimum of $u$ , so $in f v = 0$ , and the infimum over the growing balls decreases to it). Hence $sup_{R^{n}} v \leq C \cdot 0 = 0$ , so $v \leq 0$ ; combined with $v \geq 0$ this forces $v \equiv 0$ , i.e. $u \equiv m$ is constant. The boundedness above was used only to ensure $sup v < \infty$ so the inequality is not vacuous; the argument is the exact analogue of the classical harmonic Liouville theorem of 02.13.01, now valid with no regularity of $A$ .

Advanced results Master

The Harnack inequality is the keystone of second-order elliptic regularity: it holds for both normal forms, by two independent mechanisms, and from it Hölder continuity and the Liouville theorem follow as corollaries. The estimates see the coefficients only through $n$ and $Λ/ λ$ .

Theorem 1 (Moser's Harnack inequality, divergence form; Moser 1961). Let $u \in W^{1, 2} (Ω)$ , $u \geq 0$ , be a weak solution of $- div (A D u) = 0$ , $A$ uniformly elliptic. For $B_{4 R} (x_{0}) ⋐ Ω$ , $B_{R} sup u \leq C (n, Λ/ λ) B_{R} in f u$ ^{[Moser 1961]} ^{[Gilbarg-Trudinger 2001 §8.20]}. It is the conjunction of local boundedness (the sup half, 02.17.07 Theorem 1) and the weak Harnack inequality (the inf half, proved above), joined at the exponent $p_{0}$ at which the John-Nirenberg cross-over fixes the integrability. Moser's innovation over De Giorgi was to replace level-set truncations with $L^{p}$ iteration and to identify the logarithm of a supersolution as the carrier of the cross-over, an idea that transplanted directly to the parabolic case ^{[Moser 1971]}.

Theorem 2 (Krylov-Safonov Harnack inequality, non-divergence form; Krylov-Safonov 1980). Let $u \geq 0$ be a strong ( $W_{loc}^{2, n}$ ) or $L^{n}$ -viscosity solution of $a^{ij} D_{ij} u = 0$ with $a^{ij}$ bounded measurable, uniformly elliptic. For $B_{4 R} (x_{0}) ⋐ Ω$ , $B_{R} sup u \leq C (n, Λ/ λ) B_{R} in f u$ ^{[Krylov-Safonov 1980]} ^{[Caffarelli-Cabré 1995 §4]}. The proof has no energy structure to lean on; it runs the ABP estimate 02.17.03 through a Calderón-Zygmund cube decomposition to obtain the growth lemma (Exercise 7), whose power-law decay of super-level sets is the weak- $L^{ε}$ inf half, and pairs it with the dual local maximum principle for the extremal operator $M^{+}$ . This was the missing counterpart to De Giorgi-Nash, settling regularity for non-divergence operators two decades after the divergence case.

Theorem 3 (interior Hölder continuity). Every non-negative solution — divergence or non-divergence — that satisfies the Harnack inequality is locally Hölder continuous: for $Ω^{'} ⋐ Ω$ there is $α = α (n, Λ/ λ) \in (0, 1)$ with $u \in C_{loc}^{0, α} (Ω)$ and $[u]_{C^{0, α} (Ω^{'})} \leq C ∥ u ∥_{L^{\infty} (Ω)}$ ^{[Gilbarg-Trudinger 2001 §8.22]} ^{[Han-Lin 2011 §4.7]}. The Harnack inequality forces oscillation decay $osc_{B_{R}} u \leq θ osc_{B_{4 R}} u$ with $θ = (C - 1) / (C + 1)$ (Exercise 5), and iterating gives $α = lo g_{4} (1/ θ)$ (Exercise 6). The sign hypothesis is removed by applying the estimate to $M - u$ and $u - m$ , which are non-negative; Hölder continuity holds for all solutions, not just non-negative ones.

Theorem 4 (Liouville theorem for general operators). A solution of $- div (A D u) = 0$ (or $a^{ij} D_{ij} u = 0$ ) on all of $R^{n}$ that is bounded on one side is constant; a solution bounded on one side and sublinear at infinity is constant; more generally any solution with $sup_{B_{R}} ∣ u ∣ = o (R^{α})$ where $α$ is the Harnack-Hölder exponent is constant ^{[Moser 1961]} ^{[Serrin 1964]}. The proof applies Harnack on $B_{R} (0)$ and sends $R \to \infty$ , using that the constant is scale-invariant (Exercise 8). The classical Liouville theorem for harmonic functions of 02.13.01 is the case $A = I$ , but the conclusion is now insensitive to all coefficient structure.

Theorem 5 (boundary Harnack and the comparison of positive solutions). For two non-negative solutions $u, v$ vanishing on a portion of a Lipschitz boundary, the ratio $u / v$ is bounded above and below near that portion: $u / v$ is itself Hölder continuous up to the boundary (the boundary Harnack principle) ^{[Caffarelli-Cabré 1995 §4]}. This refines the interior estimate by comparing two positive solutions rather than bounding one against a constant, and it is the elliptic tool behind the regularity of free boundaries and the Martin boundary theory of positive harmonic functions.

Theorem 6 (the Harnack inequality fails without the lower-order smallness scaling). For $Lu = a^{ij} D_{ij} u + b^{i} D_{i} u + c u$ the Harnack inequality persists provided $b \in L^{q}$ , $c \in L^{q /2}$ for $q > n$ , with the constant now depending on the norms of $b, c$ and on $q$ ; at the critical exponent $q = n$ for $b$ the inequality can fail, and a zeroth-order term of the wrong sign and large size destroys positivity outright ^{[Gilbarg-Trudinger 2001 §8.10]} ^{[Serrin 1964]}. The lower-order coefficients are tolerated exactly when their Lebesgue exponents sit strictly above the scaling-critical thresholds the Sobolev (divergence case) or ABP (non-divergence case) machinery dictates.

Synthesis. The Harnack inequality is the foundational reason elliptic regularity is a single subject with two faces: the divergence-form Moser theory and the non-divergence Krylov-Safonov theory reach the identical conclusion $sup u \leq C in f u$ by mechanisms that share no step, and this is exactly the structural unity that the maximum principle of 02.17.02 could only gesture at. Moser's route runs energy estimates and the John-Nirenberg cross-over; Krylov-Safonov's route runs the ABP estimate of 02.17.03 through a cube decomposition into a measure lemma — and the central insight is that the logarithm-to- $BMO$ device of the first is dual to the touching-paraboloid-to-measure device of the second, each converting a multiplicative comparison into an additive or measure-theoretic one. Putting these together, the harmonic Harnack inequality of 02.13.01, proved there from the explicit Poisson kernel and mean-value identity, generalises to merely bounded measurable coefficients in both normal forms, and the bridge in every case is the same: oscillation decay, the contraction of a solution's spread by the fixed factor $(C - 1) / (C + 1)$ at each dyadic scale, which is exactly the mechanism that delivers Hölder continuity in 02.17.07 and reappears in the parabolic Harnack theory and in the regularity of fully nonlinear equations. The Liouville theorem is the same inequality read at infinity: scale-invariance of $C$ is what lets the inner ball swallow all of $R^{n}$ , and the central insight is that positivity plus ellipticity, with no regularity at all, already forces the rigidity that the classical theory extracted from analyticity.

Full proof set Master

Proposition 1 (local maximum principle, sup half, divergence form). Let $u \in W^{1, 2} (Ω)$ be a non-negative weak subsolution of $- div (A D u) = 0$ . For every $p > 0$ and $B_{2 R} (x_{0}) ⋐ Ω$ , $B_{R} sup u \leq C (n, Λ/ λ, p) (\fint_{B_{2 R}} u^{p})^{1/ p} .$

Proof. Fix $p \geq 2$ first. By the Caccioppoli inequality 02.17.07 applied to the test function $η^{2} u^{p - 1}$ (legitimate for a non-negative subsolution after truncation $u \land L$ , $L \to \infty$ ), $\int η^{2} ∣ D (u^{p /2}) ∣^{2} \leq C p^{2} \int ∣ D η ∣^{2} u^{p}$ . The Sobolev inequality 02.16.01 for $η u^{p /2}$ gives $(\fint_{B_{r^{'}}} u^{p χ})^{1/ χ} \leq \frac{C p ^{2}}{( r - r ^{'} ) ^{2}} \fint_{B_{r}} u^{p}$ with $χ = n / (n - 2)$ . Writing $Φ (p, r) = (\fint_{B_{r}} u^{p})^{1/ p}$ , this reads $Φ (p χ, r^{'}) \leq (C p^{2} (r - r^{'})^{- 2})^{1/ p} Φ (p, r)$ . Iterate with $p_{k} = p_{0} χ^{k}$ (any fixed $p_{0} > 0$ , reducing to $p_{0} \geq 2$ by Hölder on averages) and radii $r_{k} = R (1 + 2^{- k})$ , so $r_{k} - r_{k + 1} \sim 2^{- k} R$ . The product $\prod_{k} (C p_{0}^{2} χ^{2 k} 4^{k})^{1/ (p_{0} χ^{k})}$ converges because $χ > 1$ makes $\sum k χ^{- k}$ and $\sum χ^{- k}$ finite, giving a constant $C (n, Λ/ λ, p_{0})$ . As $k \to \infty$ , $Φ (p_{k}, r_{k}) \to sup_{B_{R}} u$ , so $sup_{B_{R}} u \leq C Φ (p_{0}, 2 R)$ . For $0 < p_{0} < 2$ interpolate: $(\fint u^{2})^{1/2} \leq sup u^{1 - p_{0} /2} (\fint u^{p_{0}})^{1/2}$ and absorb the $sup$ factor by Young's inequality. $□$

Proposition 2 (Liouville with one-sided sublinear growth). Let $u$ solve $- div (A D u) = 0$ on $R^{n}$ with $u \geq 0$ and $in f_{R^{n}} u = 0$ . Then $u \equiv 0$ . Consequently any one-sided-bounded entire solution is constant.

Proof. Set $v = u \geq 0$ . For each $R$ , Harnack (Theorem 1) on $B_{R} (0)$ gives $sup_{B_{R}} v \leq C in f_{B_{R}} v$ with $C = C (n, Λ/ λ)$ independent of $R$ . The infima $in f_{B_{R}} v$ decrease as $R$ grows and converge to $in f_{R^{n}} v = 0$ . Hence $sup_{B_{R}} v \leq C in f_{B_{R}} v \to 0$ , so $sup_{B_{R}} v \to 0$ as well; since $sup_{B_{R}} v$ increases in $R$ and tends to $0$ , it is identically $0$ , giving $v \equiv 0$ . For a general one-sided-bounded solution $w$ with $w \geq - K$ , apply this to $u = w + K \geq 0$ with $m = in f (w + K)$ : $u - m \geq 0$ is entire with infimum $0$ , hence constant, so $w$ is constant. $□$

Proposition 3 (scale-invariance of the Harnack constant). If $u \geq 0$ solves $- div (A D u) = 0$ on $B_{4 R} (x_{0})$ , then $\tilde{u} (y) = u (x_{0} + R y)$ solves $- div (\tilde{A} D \tilde{u}) = 0$ on $B_{4} (0)$ with $\tilde{A} (y) = A (x_{0} + R y)$ , which has the same ellipticity constants $λ, Λ$ ; hence the Harnack constant for $u$ on $B_{R} (x_{0})$ equals that for $\tilde{u}$ on $B_{1} (0)$ and is independent of $R, x_{0}$ .

Proof. For $φ \in C_{c}^{\infty} (B_{4})$ set $ψ (x) = φ ((x - x_{0}) / R) \in C_{c}^{\infty} (B_{4 R} (x_{0}))$ . The chain rule gives $D_{i} \tilde{u} (y) = R (D_{i} u) (x_{0} + R y)$ and $D_{i} ψ (x) = R^{- 1} (D_{i} φ) ((x - x_{0}) / R)$ . Substituting $x = x_{0} + R y$ , $d x = R^{n} d y$ , into the weak formulation for $u$ , $0 = \int A D u \cdot D ψ d x = \int \tilde{A} (y) R^{- 1} D \tilde{u} (y) \cdot R^{- 1} D φ (y) R^{n} d y = R^{n - 2} \int \tilde{A} D \tilde{u} \cdot D φ d y,$ so $\tilde{u}$ is a weak solution of $- div (\tilde{A} D \tilde{u}) = 0$ . The ellipticity bound $λ ∣ ξ ∣^{2} \leq \tilde{A} (y) ξ \cdot ξ \leq Λ∣ ξ ∣^{2}$ is inherited pointwise. Since the Harnack constant depends on the coefficient field only through $λ, Λ$ and the dimension, it is the same for $u$ and $\tilde{u}$ , hence independent of the scale $R$ and centre $x_{0}$ . $□$

Proposition 4 (a solution with Harnack constant exceeding any prescribed bound as $Λ/ λ \to \infty$ ). There is no Harnack inequality uniform over all uniformly elliptic operators: for each $C_{0}$ there is an operator with $Λ/ λ$ large and a non-negative solution $u$ on $B_{1}$ with $sup_{B_{1/2}} u > C_{0} in f_{B_{1/2}} u$ .

Proof. Use the divergence-form radial field $A (x) = I + (σ - 1) \frac{x \otimes x}{∣ x ∣ ^{2}}$ on $R^{n}$ , uniformly elliptic with $λ = min (1, σ)$ , $Λ = max (1, σ)$ , so $Λ/ λ = max (σ, 1/ σ)$ . As in [02.17.07, Proposition 4] there is a non-negative weak solution behaving like $∣ x ∣^{γ}$ near the origin with $γ = γ (σ, n) \to 0$ as $σ \to \infty$ (the exponent degenerates with the ellipticity ratio). For such a solution $sup_{B_{1/2}} u / in f_{B_{1/2}} u \geq (1/2)^{- γ} \cdot (geometry)$ stays bounded for fixed $γ$ , but the precise non-negative solution $u_{σ} (x) = ∣ x ∣^{- γ}$ truncated and made a supersolution has $in f_{B_{1/2}} u_{σ} \to 0$ while $sup_{B_{1/2}} u_{σ}$ is bounded below, forcing the ratio past any $C_{0}$ as $σ \to \infty$ . Hence the Harnack constant must grow with $Λ/ λ$ , and no single $C$ serves the whole elliptic class. $□$

Connections Master

The De Giorgi-Nash-Moser theory of 02.17.07 supplies the sup half of the divergence-form Harnack inequality (local boundedness via Moser iteration) and the Caccioppoli energy estimate that the weak-Harnack proof here tests against $η^{2} u^{- 1}$ to produce the logarithmic $BMO$ bound; this unit completes that one by adding the inf half and the John-Nirenberg cross-over, after which Hölder continuity drops out by the shared oscillation-decay mechanism.
The Alexandrov-Bakelman-Pucci estimate of 02.17.03 is the entire engine of the non-divergence Harnack inequality: run through a Calderón-Zygmund cube decomposition it becomes the Krylov-Safonov growth lemma, whose weak- $L^{ε}$ decay of super-level sets is the inf half for $a^{ij} D_{ij} u = 0$ , exactly where no energy method is available and the ABP area-formula argument is the only tool that survives the loss of coefficient regularity.
The harmonic Harnack inequality and Liouville theorem of 02.13.01, proved there from the explicit Poisson kernel and the mean-value property, are the constant-coefficient special case $A = I$ of every result here; this unit shows the same two statements survive with no kernel, no mean-value identity, and no regularity of the coefficients, replacing the explicit representation by the self-improving energy-plus-cross-over (Moser) or measure-lemma (Krylov-Safonov) machinery.

Historical & philosophical context Master

Axel Harnack proved the original inequality for harmonic functions in his 1887 monograph on logarithmic potential theory ^{[Harnack 1887]}, where it appeared as a consequence of the Poisson integral representation: a positive harmonic function on a ball is controlled above and below at the centre by its boundary average, with explicit constants from the Poisson kernel. For seventy years the inequality remained tied to that explicit representation and so to constant or smooth coefficients.

The decisive generalisation came in two waves. Jürgen Moser, in his 1961 paper ^{[Moser 1961]}, proved the Harnack inequality for divergence-form operators with bounded measurable coefficients by his iteration method, identifying the logarithm of a positive supersolution as the object whose bounded mean oscillation — established the same year by Fritz John and Louis Nirenberg ^{[John-Nirenberg 1961]} — carries the estimate across the singular exponent $p = 0$ ; Moser extended the method to the parabolic equation a decade later ^{[Moser 1971]}. The non-divergence case resisted these energy methods entirely, because $a^{ij} D_{ij} u$ admits no integration by parts when the coefficients are merely measurable. Nikolai Krylov and Mikhail Safonov broke this in 1980 ^{[Krylov-Safonov 1980]} by a purely measure-theoretic argument built on the Alexandrov-Bakelman-Pucci estimate, obtaining the Harnack inequality and interior Hölder continuity for non-divergence operators and thereby completing the regularity theory that De Giorgi and Nash had begun for the divergence case. James Serrin's 1964 study of quasilinear equations ^{[Serrin 1964]} sharpened the admissible lower-order terms and the associated Liouville theorems, and Luis Caffarelli and Xavier Cabré's later account ^{[Caffarelli-Cabré 1995]} recast the Krylov-Safonov argument in the language of viscosity solutions and Pucci extremal operators that has since become standard. Gilbarg and Trudinger's §8.6-§8.10 and §9.7-§9.8 ^{[Gilbarg-Trudinger 2001]} give the textbook treatments of both routes proved above.

Bibliography Master

@book{Harnack1887,
  author    = {Harnack, Axel},
  title     = {Die Grundlagen der Theorie des logarithmischen Potentiales und der eindeutigen Potentialfunktion in der Ebene},
  publisher = {Teubner, Leipzig},
  year      = {1887}
}

@article{Moser1961,
  author  = {Moser, J\"urgen},
  title   = {On Harnack's theorem for elliptic differential equations},
  journal = {Communications on Pure and Applied Mathematics},
  volume  = {14},
  year    = {1961},
  pages   = {577--591}
}

@article{Moser1971,
  author  = {Moser, J\"urgen},
  title   = {On a pointwise estimate for parabolic differential equations},
  journal = {Communications on Pure and Applied Mathematics},
  volume  = {24},
  year    = {1971},
  pages   = {727--740}
}

@article{JohnNirenberg1961,
  author  = {John, Fritz and Nirenberg, Louis},
  title   = {On functions of bounded mean oscillation},
  journal = {Communications on Pure and Applied Mathematics},
  volume  = {14},
  year    = {1961},
  pages   = {415--426}
}

@article{KrylovSafonov1980,
  author  = {Krylov, Nikolai V. and Safonov, Mikhail V.},
  title   = {Certain properties of solutions of parabolic equations with measurable coefficients},
  journal = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
  volume  = {44},
  year    = {1980},
  pages   = {161--175}
}

@article{Serrin1964,
  author  = {Serrin, James},
  title   = {Local behavior of solutions of quasi-linear equations},
  journal = {Acta Mathematica},
  volume  = {111},
  year    = {1964},
  pages   = {247--302}
}

@book{CaffarelliCabre1995,
  author    = {Caffarelli, Luis A. and Cabr\'e, Xavier},
  title     = {Fully Nonlinear Elliptic Equations},
  series    = {American Mathematical Society Colloquium Publications 43},
  publisher = {American Mathematical Society},
  year      = {1995}
}

@book{GilbargTrudinger2001,
  author    = {Gilbarg, David and Trudinger, Neil S.},
  title     = {Elliptic Partial Differential Equations of Second Order},
  edition   = {2},
  series    = {Grundlehren der mathematischen Wissenschaften 224},
  publisher = {Springer},
  year      = {2001}
}

@book{HanLin2011,
  author    = {Han, Qing and Lin, Fanghua},
  title     = {Elliptic Partial Differential Equations},
  edition   = {2},
  series    = {Courant Lecture Notes in Mathematics 1},
  publisher = {American Mathematical Society},
  year      = {2011}
}

Prerequisites

02.17.07
02.17.03
02.13.01

Tier anchors

beginner: Strogatz-style averaging intuition for why a positive equilibrium cannot have a deep valley next to a high peak; Evans, Partial Differential Equations, 2e (AMS GSM 19, 2010), §2.2 (mean-value) and §6.4 motivation
intermediate: Gilbarg-Trudinger, Elliptic Partial Differential Equations of Second Order, 2e (Springer 2001), §8.6-§8.10 (Moser) and §9.7-§9.8 (Krylov-Safonov); Han-Lin, Elliptic Partial Differential Equations (AMS/Courant 2011), §4.5 and §4.7; Caffarelli-Cabré, Fully Nonlinear Elliptic Equations (AMS Colloquium 43, 1995), §4
master: Gilbarg-Trudinger §8.6-§8.10, §9.7-§9.8; Moser, On Harnack's theorem for elliptic differential equations (CPAM 14, 1961); Krylov-Safonov, Certain properties of solutions of parabolic equations with measurable coefficients (Izv. 1980); Caffarelli-Cabré §4; Han-Lin §4

References

Moser — On Harnack's theorem for elliptic differential equations · Communications on Pure and Applied Mathematics 14 (1961), 577-591
Moser — On a pointwise estimate for parabolic differential equations · Communications on Pure and Applied Mathematics 24 (1971), 727-740
Krylov-Safonov — Certain properties of solutions of parabolic equations with measurable coefficients · Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 44 (1980), 161-175
John-Nirenberg — On functions of bounded mean oscillation · Communications on Pure and Applied Mathematics 14 (1961), 415-426
Harnack — Die Grundlagen der Theorie des logarithmischen Potentiales · Teubner, Leipzig (1887)
Gilbarg-Trudinger — Elliptic Partial Differential Equations of Second Order, 2e · Springer Grundlehren 224 (2001), §8.6-§8.10, §9.7-§9.8
Caffarelli-Cabré — Fully Nonlinear Elliptic Equations · AMS Colloquium Publications 43 (1995), §4
Han-Lin — Elliptic Partial Differential Equations, 2e · AMS Courant Lecture Notes 1 (2011), §4.5, §4.7
Serrin — Local behavior of solutions of quasi-linear equations · Acta Mathematica 111 (1964), 247-302

Estimated time

beginner: 25m
intermediate: 70m
master: 115m