Moser's weak Harnack inequality and the Harnack inequality #
Let a be measurable and uniformly elliptic on Ω ⊆ ℝⁿ with constants 0 < λ ≤ Λ. This file
proves, in every dimension n ≥ 1, the two Harnack inequalities of Moser for the divergence-form
equation -∂ᵢ(aⁱʲ ∂ⱼu) = 0, with no regularity of the coefficients beyond measurability.
The weak Harnack inequality: if
u ∈ H¹(Ω)is a weak supersolution,a(u, v) ≥ 0for every nonnegativev ∈ H¹₀(Ω), andu ≥ 0almost everywhere, then for every exponents > 0with(n - 2) s < n(that iss < n/(n - 2)whenn ≥ 3, and anys > 0whenn ≤ 2) and every ball withB(x₀, 8√n r) ⊆ Ω,(⨍_{B(x₀, r)} u^s)^{1/s} ≤ C u(x)for almost everyx ∈ B(x₀, r),that is, the
L^saverage ofuis controlled by its essential infimum. In particularu^sis integrable onB(x₀, r), which is part of the conclusion.The Harnack inequality: if
uis a weak solution,a(u, v) = 0for everyv ∈ H¹₀(Ω), andu ≥ 0almost everywhere, then on every ball withB(x₀, 16√n r) ⊆ Ω,u(y) ≤ C u(x)for almost everyx, y ∈ B(x₀, r),that is,
ess sup_{B(x₀, r)} u ≤ C ess inf_{B(x₀, r)} u.
In both, C depends only on λ, Λ, the dimension, the normalization of the Haar measure and,
for the weak Harnack inequality, the exponent s. It does not depend on u or on the ball, so the
estimates are scale invariant.
The weak Harnack inequality at a small exponent is first proved for supersolutions with
u ≥ ε > 0, with a constant independent of ε, as in Moser's proof. On B = B(x₀, 2r), at the
exponent p₀ = min(cλ/Λ, 2) of the crossover estimate, it combines
- the lower bound from negative powers,
u⁻¹ ≤ D |B|^{-1/p₀} ‖u⁻¹‖_{L^{p₀}(B)}onB(x₀, r)(TauCeti.PDE.exists_ae_inv_value_le_mul_rpow_mul_setIntegral_rpow), with - the crossover estimate
⨍_B u^{p₀} · ⨍_B u^{-p₀} ≤ K, from the John–Nirenberg inequality forlog u(TauCeti.PDE.exists_setAverage_rpow_mul_setAverage_rpow_neg_le).
Together they bound (⨍_B u^{p₀})^{1/p₀} by C u(x) for almost every x ∈ B(x₀, r), in every
dimension n ≥ 1. A nonnegative supersolution is reduced to this case by applying it to u + δ
on the bounded ball B(x₀, 8√n r), where constants lie in H¹, and letting δ → 0. Moser's
iteration for small positive powers of nonnegative supersolutions
(TauCeti.PDE.exists_setAverage_rpow_rpow_inv_le_mul_setAverage_rpow_rpow_inv) then raises the
exponent from p₀ to any s with (n - 2) s < n. The Harnack inequality combines the weak
Harnack inequality at s = 1, which is admissible in every dimension, with the local boundedness
of subsolutions in L¹ (TauCeti.PDE.exists_ae_value_le_add_mul_rpow_mul_setIntegral_rpow).
Main declarations #
TauCeti.PDE.exists_pos_ae_setAverage_rpow_rpow_inv_le_mul_value: the weak Harnack inequality at a small exponentp₀, in every dimension, for nonnegative supersolutions.TauCeti.PDE.exists_ae_setAverage_rpow_rpow_inv_le_mul_value: Moser's weak Harnack inequality for nonnegative supersolutions, for every exponents > 0with(n - 2) s < n.TauCeti.PDE.exists_ae_ae_value_le_mul_value: Moser's Harnack inequality for nonnegative solutions.
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, Theorems 8.17, 8.18 and 8.20.
Moser's weak Harnack inequality at a small exponent. Let n ≥ 1 be the dimension. There
are an exponent p₀ ∈ (0, 2] and a constant C > 0, depending only on λ, Λ, the dimension
and the normalization of the additive Haar measure mu, with the following property. Let a be
measurable and uniformly elliptic on Ω with constants λ, Λ, and let u ∈ H¹(Ω) be a
nonnegative weak supersolution of -∂ᵢ(aⁱʲ ∂ⱼu) ≥ 0, that is a(u, v) ≥ 0 for every
nonnegative v ∈ H¹₀(Ω). Then on every ball with B(x₀, 8√n r) ⊆ Ω,
(⨍_{B(x₀, 2r)} u^{p₀})^{1/p₀} ≤ C u(x) for almost every x ∈ B(x₀, r).
The constant does not depend on u or the ball.
Moser's weak Harnack inequality. Let n ≥ 1 be the dimension. For every exponent s > 0
with (n - 2) s < n, that is s < n/(n - 2) when n ≥ 3 and s arbitrary when n ≤ 2, there
is C > 0, depending only on λ, Λ, s, the dimension and the normalization of the additive
Haar measure mu, with the following property. Let a be measurable and uniformly elliptic on
Ω with constants λ, Λ, and let u ∈ H¹(Ω) be a nonnegative weak supersolution of
-∂ᵢ(aⁱʲ ∂ⱼu) ≥ 0, that is a(u, v) ≥ 0 for every nonnegative v ∈ H¹₀(Ω). Then on every ball
with B(x₀, 8√n r) ⊆ Ω,
(⨍_{B(x₀, r)} u^s)^{1/s} ≤ C u(x) for almost every x ∈ B(x₀, r),
and u^s is integrable on B(x₀, r), so the left side is a genuine average even for s > 2.
The constant does not depend on u or the ball.
Moser's Harnack inequality. Let n ≥ 1 be the dimension. There is C > 0, depending
only on λ, Λ, the dimension and the normalization of the additive Haar measure mu, with the
following property. Let a be measurable and uniformly elliptic on Ω with constants λ, Λ,
and let u ∈ H¹(Ω) be a weak solution of -∂ᵢ(aⁱʲ ∂ⱼu) = 0, that is a(u, v) = 0 for every
v ∈ H¹₀(Ω), with u ≥ 0 almost everywhere. Then on every ball with B(x₀, 16√n r) ⊆ Ω,
u(y) ≤ C u(x) for almost every x, y ∈ B(x₀, r),
that is, ess sup_{B(x₀, r)} u ≤ C ess inf_{B(x₀, r)} u. The constant does not depend on u
or the ball.