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The Harnack estimate needs an additive forcing term

Statement refuted

Statement refuted. For every $n \geq 1$ , every nonnegative weak solution $u \in H^{1} (B_{1} (0) \subset R^{n})$ of $- Δ u = f$ with $f \in L^{\infty} (B_{1} (0))$ satisfies the forcing-free comparison $\sup_{B_{1 / 2} (0)} u \leq C \inf_{B_{1 / 2} (0)} u$ with a constant $C$ independent of $f$ and $u$ .

Counterexample. For any $n \geq 1$ , define $u (x) = ∣ x ∣^{2} / (2 n)$ and $f \equiv - 1$ on $Ω = B_{3} (0) \subset R^{n}$ , and restrict them to $B_{1} (0)$ for the refuted estimate. Then $- Δ u = f$ classically, $u \geq 0$ , and on $B_{1 / 2} (0)$ the infimum is $0$ while the supremum $1 / (8 n) > 0$ is approached as $∣ x ∣ ↑ 1 / 2$ . Hence the forcing-free comparison fails for every constant $C$ ; the additive term in Harnack inequality for nonnegative weak solutions and Weak Harnack inequality for nonnegative supersolutions cannot be omitted.

Facts & Assumptions

Given: The Axiom of Choice and Countable Choice; an integer $n \geq 1$ ; $Ω = B_{3} (0) \subset R^{n}$ ; the function $u (x) = ∣ x ∣^{2} / (2 n)$ ; and the constant source $f \equiv - 1$ .

[F1]

Elementary differentiation: $D_{i} u = x_{i} / n$ and $Δ u = 2 n / (2 n) = 1$ , so $- Δ u = - 1 = f$ classically on $B_{3} (0)$ ; the Laplacian is the operator $- Δ$ with $a^{i j} = δ^{i j}$ , $b = c = 0$ in the convention of Uniformly elliptic divergence-form operators and their sesquilinear forms (Local weak solutions of a divergence-form operator).

[F2]

On the open half-ball $B_{1 / 2} (0)$ , $u$ has infimum $0$ attained at the origin, while its supremum $1 / (8 n)$ is approached along $x_{j} = (1 / 2 - 1 / j) e_{0}$ for $j > 2$ and is not attained; continuity makes this equal to the essential supremum (The average of a locally integrable function over a Euclidean ball, The essential supremum of a measurable function with respect to a measure).

[F3]

For $n \geq 2$ , the displayed Harnack statements carry the forcing additively: ${ess sup}_{B_{R / 2}} u \leq C ({ess inf}_{B_{R / 2}} u + R^{2 - n / q} ∥ F ∥_{L^{q} (B_{2 R})})$ for weak solutions of $L_{0} u = - F$ , and the analogous bound with the same additive structure holds for nonnegative supersolutions (Harnack inequality for nonnegative weak solutions, Weak Harnack inequality for nonnegative supersolutions, Weak subsolutions and supersolutions of a divergence-form equation). The polynomial counterexample itself is valid in every $n \geq 1$ by [F1]-[F2].

Counterexample

1.1givenF1

The function solves the equation and is nonnegative. By [F1], $u$ is smooth on $B_{3} (0)$ with $- Δ u = - 1 = f$ , so its restriction to $B_{1} (0)$ is a weak solution in the local sense; $u \geq 0$ and $f \equiv - 1$ is bounded.

2.1step 1.1F2

The extrema on the half ball. By [F2], $\inf_{B_{1 / 2} (0)} u = 0$ and $\sup_{B_{1 / 2} (0)} u = 1 / (8 n) > 0$ ; hence for every real constant $C$ one has $\sup_{B_{1 / 2} (0)} u = 1 / (8 n) > 0 = C \cdot 0 = C \inf_{B_{1 / 2} (0)} u$ , so the forcing-free comparison $\sup \leq C \inf$ fails for every $C$ .

3.1step 2.1F3∎

For $n \geq 2$ , the additive term repairs the estimate and cannot be dropped. With $F = - f = 1$ , the solution is defined on $Ω = B_{3} (0)$ , so the doubled ball $B_{2} (0) ⋐ Ω$ is admissible in [F3]. Its source norm is $∥ F ∥_{L^{q} (B_{2})} = ∣ B_{2} ∣^{1 / q}$ , and the estimate with $R = 1$ has the nonzero additive term $∣ B_{2} ∣^{1 / q}$ ; the polynomial still has zero infimum and positive supremum on $B_{1 / 2}$ . Thus no finite constant can replace the additive term by $C \inf$ . Steps 1.1–2.1 already verify the forcing-free failure for every $n \geq 1$ , independently of invoking [F3]. All verifications use the explicit polynomial and the cited statements, with no choice principle beyond the declared Axiom of Choice and Countable Choice.

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Sources

Brian Krummel, DeGiorgi-Nash lecture notes (15 March 2016; complete 9-page notes) (standard reference, not scraped)
Leon Simon, Lectures on Partial Differential Equations (Stanford University; complete author scan, 118 sheets reproducing the 223 printed pages of the manuscript, two logical pages per sheet) (standard reference, not scraped)