Moser's iteration for small positive powers of a 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 -∂ᵢ(aⁱʲ ∂ⱼu) ≥ 0, bounded below by a constant ε > 0.
Suppose that W^{1,2}_0(Ω) satisfies a Sobolev inequality ‖v‖_q ≤ S ‖∇v‖₂. For every p < 1,
the power u^{p/2} then satisfies a reverse Hölder inequality on concentric balls
B(x₀, r) ⊂ B(x₀, ρ) ⊆ Ω:
‖u^{p/2}‖_{L^q(B(x₀, r))} ≤ S C (1 + |p| Λ / ((1 - p) λ)) / (ρ - r) · ‖u^{p/2}‖_{L²(B(x₀, ρ))},
with C depending only on the dimension. For 0 < p < 1 and χ = q/2 this bounds the L^{χp}
norm of u on the smaller ball by its L^p norm on the larger one. Iterating it finitely often,
along the exponents p₀ χ^k and radii shrinking from R to R/2, gives Moser's estimate for
small positive powers: in every dimension n ≥ 1, for all p > 0 and s > 0 with
(n - 2) s < n,
‖u‖_{L^s(B(x₀, R/2))} ≤ D R^{n/s - n/p} ‖u‖_{L^p(B(x₀, R))}.
The condition on s reads s < n/(n - 2) for n ≥ 3 and is void for n ≤ 2. This is the half
of Moser's weak Harnack inequality that passes from a small power p, reached from negative
powers through the John–Nirenberg inequality for log u, up to any such power s.
The iteration is run on the ball B(x₀, R) itself, with the Sobolev inequality of
W^{1,2}_0(B(x₀, R)) for sets of finite measure
(TauCeti.W1p.eLpNorm_value_le_mul_rpow_mul_enorm_gradient_of_eq_ball) at an exponent q = 2χ
with χ > s and 1/2 - 1/q < 1/n; such a q exists exactly when (n - 2) s < n. Its constant
is a multiple of R ^ (1 - n (1/2 - 1/q)), and this power of R is what makes the estimate scale
invariant. In dimension n ≥ 3 the Gagliardo–Nirenberg–Sobolev inequality would serve for
s < n/(n - 2) as well, but in dimensions 1 and 2 there is no critical exponent and the
dependence of the constant on the ball is essential. The restriction s < χ keeps every exponent
p₀ χ^k used in the iteration below 1, where the Caccioppoli inequality for powers holds. Its
scale-invariant averaged form, for p ≤ 2, extends to nonnegative supersolutions: apply it to
u + δ on the ball, where constants lie in H¹, and let δ → 0 by dominated convergence.
The reverse Hölder inequality applies the Sobolev inequality to ψ u^{p/2}, for a cutoff ψ equal
to 1 on B(x₀, r), and bounds its gradient by the Caccioppoli inequality for powers, in its
form for u^{p/2}
(TauCeti.PDE.UniformlyEllipticOn.setIntegral_sq_mul_norm_gradient_sq_le_of_ae_eq_rpow).
Main declarations #
TauCeti.PDE.exists_eLpNorm_rpow_le_mul_eLpNorm_rpow: the reverse Hölder inequality for powers of a positive supersolution, under a Sobolev inequality onW^{1,2}_0(Ω).TauCeti.PDE.exists_eLpNorm_le_ofReal_rpow_mul_eLpNorm: its form for0 < p < 1as a bound of theL^{χp}norm ofuby itsL^pnorm.TauCeti.PDE.exists_eLpNorm_le_mul_rpow_mul_eLpNorm: Moser's estimate for small positive powers, in every dimension.TauCeti.PDE.exists_setAverage_rpow_rpow_inv_le_mul_setAverage_rpow_rpow_inv: its scale-invariant form(⨍_{B(x₀, R/2)} u^s)^{1/s} ≤ D (⨍_{B(x₀, R)} u^p)^{1/p}forp ≤ 2, together with the integrability ofu^sonB(x₀, R/2), for nonnegative supersolutions.
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.
Moser's reverse Hölder inequality for powers of a positive supersolution. 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 suppose that
W^{1,2}_0(Ω) satisfies the Sobolev inequality ‖v‖_q ≤ S ‖∇v‖₂. 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 p < 1 and all
concentric balls B(x₀, r) ⊂ B(x₀, ρ) ⊆ Ω,
‖u^{p/2}‖_{L^q(B(x₀, r))} ≤ S C (1 + |p| Λ / ((1 - p) λ)) / (ρ - r) · ‖u^{p/2}‖_{L²(B(x₀, ρ))}.
The constant depends on neither u nor ε.
One step of Moser's iteration for small positive powers. There is a constant C ≥ 0,
depending only on the dimension, with the following property. Under the hypotheses of
TauCeti.PDE.exists_eLpNorm_rpow_le_mul_eLpNorm_rpow, with a finite Sobolev exponent q,
write χ = q / 2.
Then for every 0 < p < 1 and all concentric balls B(x₀, r) ⊂ B(x₀, ρ) ⊆ Ω,
‖u‖_{L^{χp}(B(x₀, r))} ≤ (S C (1 + p Λ / ((1 - p) λ)) / (ρ - r))^{2/p} ‖u‖_{L^p(B(x₀, ρ))}.
For q > 2 the exponent χp exceeds p, so the integrability of u improves on the smaller
ball; the constant blows up as p → 1.
Moser's iteration for small positive powers of a supersolution. Let n ≥ 1 be the
dimension, and let p > 0 and s > 0 with (n - 2) s < n, that is s < n/(n - 2) when n ≥ 3
and s arbitrary when n ≤ 2. There is D > 0, depending on λ, Λ, p, 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 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 on every ball B(x₀, R) ⊆ Ω,
‖u‖_{L^s(B(x₀, R/2))} ≤ D R^{n/s - n/p} ‖u‖_{L^p(B(x₀, R))}.
Since |B(x₀, R)| is proportional to Rⁿ, this says
(⨍_{B(x₀, R/2)} u^s)^{1/s} ≤ D' (⨍_{B(x₀, R)} u^p)^{1/p} with D' independent of R. The
constant does not depend on u or ε.
Moser's iteration for small positive powers, averaged form. Let n ≥ 1 be the
dimension. For exponents 0 < p ≤ 2 and s > 0 with (n - 2) s < n, that is s < n/(n - 2)
when n ≥ 3 and s arbitrary when n ≤ 2, there is D > 0, depending on λ, Λ, p, 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. Then on every ball
B(x₀, R) ⊆ Ω,
(⨍_{B(x₀, R/2)} u^s)^{1/s} ≤ D (⨍_{B(x₀, R)} u^p)^{1/p},
and u^s is integrable on B(x₀, R/2), so the left side is a genuine average even for s > 2.
The constant does not depend on u or the ball. The bound p ≤ 2 makes u^p integrable on
B(x₀, R), since u ∈ L²(Ω).