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Scaled Caccioppoli inequality on concentric balls

Statement

In the setting of The Caccioppoli inequality for weak elliptic solutions, for concentric balls $B_{r} (x_{0}) ⋐ B_{R} (x_{0}) ⋐ Ω$ one has, with the explicit scale, $∥ D u ∥_{L^{2} (B_{r} (x_{0}))}^{2} \leq C (\frac{1}{(R - r)^{2}} ∥ u ∥_{L^{2} (B_{R} (x_{0}))}^{2} + ∥ u ∥_{L^{2} (B_{R} (x_{0}))}^{2} + ∥ f ∥_{L^{2} (B_{R} (x_{0}))}^{2}),$ where $C = C (n, θ, M_{a}, M_{b}, M_{c})$ does not depend on $r, R$ or on $x_{0}$ . The displayed $(R - r)^{- 2}$ is the scale used in the nested-ball iteration; the constants are not asserted to be sharp, and no claim is made as $r \to R$ .

Facts & Assumptions

Given: Countable Choice; the setting of The Caccioppoli inequality for weak elliptic solutions: an open set $Ω \subseteq R^{n}$ , a scalar field $K$ , coefficients $a^{i j}, b^{i}, c$ with ellipticity constant $θ$ and bounds $M_{a}, M_{b}, M_{c}$ , a class $f \in L_{l o c}^{2} (Ω)$ and a local weak solution $u \in H^{1} (Ω)$ of $L u = f$ ; and concentric balls $B_{r} (x_{0}) ⋐ B_{R} (x_{0}) ⋐ Ω$ .

[F1]

Caccioppoli estimate: for every ball $B_{R} (x_{0}) ⋐ Ω$ and every $0 < r < R$ there is $C = C (n, θ, M_{a}, M_{b}, M_{c})$ with $\int_{B_{r} (x_{0})} ∣ D u ∣^{2} d x \leq C (\frac{1}{(R - r)^{2}} \int_{B_{R} (x_{0})} ∣ u ∣^{2} d x + \int_{B_{R} (x_{0})} ∣ u ∣^{2} d x + \int_{B_{R} (x_{0})} ∣ f ∣^{2} d x) .$ (The Caccioppoli inequality for weak elliptic solutions)

[F2]

$L^{2}$ norms of classes are $∥ w ∥_{L^{2} (A)} = (\int_{A} ∣ w ∣^{2} d x)^{1 / 2}$ , so each integral in [F1] is the square of the corresponding $L^{2}$ norm. (The space $L^{p} (μ)$ as the quotient by null functions)

Proof

technique · direct

1.1F1given

The hypotheses of [F1] are exactly those of the given setting, so [F1] provides a constant $C = C (n, θ, M_{a}, M_{b}, M_{c})$ with $\int_{B_{r} (x_{0})} ∣ D u ∣^{2} d x \leq C (\frac{1}{(R - r)^{2}} \int_{B_{R} (x_{0})} ∣ u ∣^{2} d x + \int_{B_{R} (x_{0})} ∣ u ∣^{2} d x + \int_{B_{R} (x_{0})} ∣ f ∣^{2} d x) .$ The constant does not depend on $x_{0}, r, R$ because [F1] itself manufactures it only from $n, θ, M_{a}, M_{b}, M_{c}$ .

2.1F2step 1.1algebra∎

Writing each integral as the square of the $L^{2}$ norm via [F2], the inequality of step 1.1 becomes exactly the displayed estimate: the left side is $∥ D u ∥_{L^{2} (B_{r} (x_{0}))}^{2}$ and the right side is $C ((R - r)^{- 2} ∥ u ∥_{L^{2} (B_{R} (x_{0}))}^{2} + ∥ u ∥_{L^{2} (B_{R} (x_{0}))}^{2} + ∥ f ∥_{L^{2} (B_{R} (x_{0}))}^{2})$ . Nothing was changed except the notation, so the scale $(R - r)^{- 2}$ and the independence of the constant from $r, R, x_{0}$ hold as asserted.

Source notes

Simon's Lemma 1 (printed p. 59) records the constant $C (M, θ, ρ, R, n)$ for the estimate; Hunter's (4.39) (printed p. 112) uses the same scale. The zero-order term is retained explicitly because it cannot be absorbed into the $(R - r)^{- 2}$ term when $R - r \geq 1$ ; it is controlled at the base of every nested-ball iteration.

Depends on

Used by

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Sources

Leon Simon, Lectures on Partial Differential Equations (Stanford, complete 223-page author scan) (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (UC Davis, revised 18 June 2014, complete 242-page two-quarter graduate notes) (standard reference, not scraped)