The logarithmic Caccioppoli inequality for positive supersolutions #
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). If u is bounded below by a positive
constant on the support of a smooth cutoff ψ compactly supported in Ω, then
∫_Ω ψ² ‖∇u‖² / u² ≤ (2Λ/λ)² ∫_Ω ‖∇ψ‖².
This is the logarithmic Caccioppoli inequality: ‖∇u‖ / u = ‖∇ log u‖, so the gradient of
log u is controlled by the cutoff alone, independently of the size of u. Combined with the
Poincaré–Wirtinger inequality on balls it shows that log u has bounded mean oscillation: on
every ball B(x₀, R) with B(x₀, 2R) ⊆ Ω, if u ≥ m almost everywhere on the doubled ball
B(x₀, 2R) for some constant m > 0, then
∫_{B(x₀, R)} (log u - (log u)_{B(x₀, R)})² ≤ C (Λ/λ)² |B(x₀, R)|,
with C depending only on the dimension. This estimate is the bridge between the bounds for
subsolutions and for supersolutions in Moser's proof of the Harnack inequality: through the
John–Nirenberg inequality, or the Bombieri–Giusti lemma, it is what allows the averages of a
small positive power of u and of its reciprocal to be compared.
The proof tests the supersolution inequality against ψ²/u, which lies in H¹₀(Ω), with weak
gradient 2ψ ∇ψ / u - ψ² ∇u / u². Since u is only bounded below on the support of ψ, the
reciprocal is realised as a composition F ∘ u with a C¹ function F that has bounded
derivative, vanishes near 0 and agrees with t ↦ 1/t for t ≥ m/2; the logarithm is handled
in the same way for the mean-oscillation bound.
Main declarations #
TauCeti.PDE.UniformlyEllipticOn.setIntegral_sq_mul_norm_gradient_sq_div_sq_le: the logarithmic Caccioppoli inequality.TauCeti.PDE.exists_setIntegral_ball_norm_gradient_sq_div_sq_le: its form on concentric balls,∫_{B(x₀, r)} ‖∇u‖² / u² ≤ (2Λ/λ)² (c / (R - r))² |B(x₀, R)|.TauCeti.PDE.exists_setIntegral_ball_log_sub_setAverage_sq_le: theL²mean oscillation oflog uon a ball is bounded by a dimensional constant times(Λ/λ)².
References #
- J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961).
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,
§8.6 (the estimate for
log uin the proof of Theorem 8.18). - Q. Han, F. Lin, Elliptic Partial Differential Equations, Chapter 4.
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). Let ψ be smooth
and compactly supported in Ω, and suppose that u ≥ m almost everywhere on the support of ψ
for some m > 0. Then
∫_Ω ψ² ‖∇u‖² / u² ≤ (2Λ/λ)² ∫_Ω ‖∇ψ‖².
The left side is ∫_Ω ψ² ‖∇ log u‖²; the bound does not depend on m, on the size of u, or
on any regularity of the coefficients beyond measurability.
The logarithmic Caccioppoli inequality on concentric balls. There is a constant c > 0,
depending only on the dimension, such that the following holds for every additive Haar measure
μ. 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. If B(x₀, r) ⊆ B(x₀, R) ⊆ Ω with
0 < r < R, and u ≥ m almost everywhere on B(x₀, R) for some m > 0, then
∫_{B(x₀, r)} ‖∇u‖² / u² ≤ (2Λ/λ)² (c / (R - r))² |B(x₀, R)|.
Bounded mean oscillation of the logarithm of a positive supersolution (Moser). There is
a constant C > 0, depending only on the dimension, such that the following holds for every
additive Haar measure μ. 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. If
B(x₀, 2R) ⊆ Ω and u ≥ m almost everywhere on B(x₀, 2R) for some m > 0, then
∫_{B(x₀, R)} (log u - (log u)_{B(x₀, R)})² ≤ C (Λ/λ)² |B(x₀, R)|,
where (log u)_{B(x₀, R)} is the mean of log u over the ball. The bound does not depend on
m, on R, or on the size of u.