The logarithm of a positive weak supersolution #
Let a be measurable and uniformly elliptic on Ω ⊆ ℝⁿ with constants 0 < λ ≤ Λ, and let
u ∈ H¹(Ω) be a weak supersolution of the divergence-form equation
-∂ᵢ(aⁱʲ ∂ⱼu) ≥ 0 in Ω,
meaning a(u, v) ≥ 0 for every nonnegative v ∈ H¹₀(Ω), which is bounded below by a constant
ε > 0. Then log u is a Sobolev function with weak gradient ∇u / u
(TauCeti.W1p.exists_value_gradient_ae_eq_comp_of_le), and this file shows that this gradient is
controlled by the ellipticity constants alone:
the logarithmic Caccioppoli inequality: for every smooth
ψcompactly supported inΩ,∫_Ω ψ² ‖∇u‖² / u² ≤ (2Λ/λ)² ∫_Ω ‖∇ψ‖².Unlike the Caccioppoli inequality for
uitself, the right side does not involveuat all;on every ball with
B(x₀, 2r) ⊆ Ω, the scale-invariant bound∫_{B(x₀, r)} ‖∇u‖² / u² ≤ C (Λ/λ)² |B(x₀, r)| / r²;consequently, by the Poincaré–Wirtinger inequality on balls,
log uhas bounded mean oscillation:⨍_{B(x₀, r)} |log u - (log u)_{B(x₀, r)}| ≤ C Λ/λon every such ball.
The constants C depend only on the dimension; nothing depends on u or on the lower bound ε,
which serves only to make u⁻¹ and log u Sobolev functions. No regularity of the coefficients
beyond measurability is used.
The mean oscillation bound is the hypothesis of the John–Nirenberg inequality. In Moser's proof of
the weak Harnack inequality, John–Nirenberg turns it into the crossover estimate
⨍_B u^p · ⨍_B u^{-p} ≤ C for a small p > 0, which links bounds on negative powers of a
supersolution to bounds on positive ones.
The logarithmic Caccioppoli inequality is the case p = 0 of the Caccioppoli inequality for
powers of u
(TauCeti.PDE.UniformlyEllipticOn.setIntegral_sq_mul_rpow_mul_norm_gradient_sq_le), which tests
the supersolution inequality against ψ² u^{p-1}, here ψ² u⁻¹.
Main declarations #
TauCeti.PDE.UniformlyEllipticOn.setIntegral_sq_mul_norm_gradient_div_value_sq_le: the logarithmic Caccioppoli inequality.TauCeti.PDE.exists_setIntegral_ball_norm_gradient_div_value_sq_le: the bound for∫ ‖∇ log u‖²on balls.TauCeti.PDE.exists_setAverage_abs_log_sub_setAverage_le:log uhas bounded mean oscillation.
References #
- J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577–591.
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, §8.6 (the proof of Theorem 8.18).
The logarithmic Caccioppoli inequality. Let a be measurable and uniformly elliptic on
Ω with constants 0 < λ ≤ Λ, and let u ∈ H¹(Ω) be a weak supersolution of
-∂ᵢ(aⁱʲ ∂ⱼu) ≥ 0, that is a(u, v) ≥ 0 for every nonnegative v ∈ H¹₀(Ω), with u ≥ ε
almost everywhere for some ε > 0. Then for every smooth ψ compactly supported in Ω,
∫_Ω ψ² (‖∇u‖ / u)² ≤ (2Λ/λ)² ∫_Ω ‖∇ψ‖².
The integrand on the left is ψ² ‖∇ log u‖², and the right side depends on neither u nor
ε.
The gradient of log u on balls. There is a constant C ≥ 0, depending only on the
dimension, with the following property. Let a be measurable and uniformly elliptic on Ω with
constants 0 < λ ≤ Λ, and let u ∈ H¹(Ω) be a weak supersolution of -∂ᵢ(aⁱʲ ∂ⱼu) ≥ 0 with
u ≥ ε almost everywhere for some ε > 0. Then on every ball with B(x₀, 2r) ⊆ Ω,
∫_{B(x₀, r)} (‖∇u‖ / u)² ≤ C (Λ/λ)² |B(x₀, r)| / r².
So the average of ‖∇ log u‖² over B(x₀, r) is at most C (Λ/λ)² r⁻², the power of r
dictated by scaling.
The logarithm of a positive supersolution has bounded mean oscillation. There is a
constant C ≥ 0, depending only on the dimension, with the following property. Let a be
measurable and uniformly elliptic on Ω with constants 0 < λ ≤ Λ, and let u ∈ H¹(Ω) be a
weak supersolution of -∂ᵢ(aⁱʲ ∂ⱼu) ≥ 0 with u ≥ ε almost everywhere for some ε > 0. Then on
every ball with B(x₀, 2r) ⊆ Ω,
⨍_{B(x₀, r)} |log u - ⨍_{B(x₀, r)} log u| ≤ C Λ/λ.
The bound is uniform over all such balls, and independent of u and ε: this is the hypothesis
of the John–Nirenberg inequality for log u.