The Caccioppoli inequality for powers of a positive 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. For every exponent p < 1 and every smooth ψ compactly supported in Ω, this file
proves
∫_Ω ψ² u^{p-2} ‖∇u‖² ≤ (2Λ / ((1 - p) λ))² ∫_Ω ‖∇ψ‖² u^p.
For p ≠ 0 the left side is (2/p)² ∫_Ω ψ² ‖∇(u^{p/2})‖², so this is a Caccioppoli
inequality for the power u^{p/2}; for p = 0 it bounds ∫_Ω ψ² ‖∇ log u‖². These are the
energy estimates of Moser's iteration: the case 0 < p < 1 feeds the reverse Hölder inequality
for small positive powers of a supersolution, the case p < 0 the bound for its negative powers,
and the case p = 0 the bounded mean oscillation of log u. The constant depends only on λ,
Λ and p, and blows up as p → 1; no regularity of the coefficients beyond measurability is
used, and the lower bound ε serves only to make the powers of u Sobolev functions.
The inequality is proved by testing the supersolution inequality against ψ² u^{p-1}, which is
nonnegative and lies in H¹₀(Ω). Its gradient is (p - 1) ψ² u^{p-2} ∇u + 2 ψ u^{p-1} ∇ψ, and
the coefficient p - 1 < 0 of the first term turns the supersolution inequality into an upper
bound for ∫ ψ² u^{p-2} ⟨a ∇u, ∇u⟩; Young's inequality absorbs the cross term.
Main declarations #
TauCeti.PDE.UniformlyEllipticOn.setIntegral_sq_mul_rpow_mul_norm_gradient_sq_le: the Caccioppoli inequality for powers of a positive supersolution.TauCeti.PDE.UniformlyEllipticOn.setIntegral_sq_mul_norm_gradient_sq_le_of_ae_eq_rpow: its form as a Caccioppoli inequality for the poweru^{p/2}.
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 Caccioppoli inequality for powers of a positive supersolution. 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 p < 1 and every
smooth ψ compactly supported in Ω,
∫_Ω ψ² u^{p-2} ‖∇u‖² ≤ (2Λ / ((1 - p) λ))² ∫_Ω ‖∇ψ‖² u^p.
For p ≠ 0 the integrand on the left is (2/p)² ψ² ‖∇(u^{p/2})‖²; for p = 0 it is
ψ² ‖∇ log u‖². The constant depends on neither u nor ε.
The Caccioppoli inequality for the power w = u^{p/2} of a positive supersolution. Under
the hypotheses of TauCeti.PDE.UniformlyEllipticOn.setIntegral_sq_mul_rpow_mul_norm_gradient_sq_le,
let w ∈ H¹(Ω) have value u^{p/2} and gradient (p/2) u^{p/2-1} ∇u, as provided by
TauCeti.W1p.exists_value_gradient_ae_eq_rpow. Then for every p < 1 and every smooth ψ
compactly supported in Ω,
∫_Ω ψ² ‖∇w‖² ≤ (|p| Λ / ((1 - p) λ))² ∫_Ω ‖∇ψ‖² w².
This is the energy estimate that Moser's iteration combines with a Sobolev inequality for ψ w;
at p = 0 both sides vanish, since w is then constant.