The Caccioppoli inequality for truncations with lower-order terms #
Let u ∈ H¹(Ω) be a weak subsolution of the full divergence-form equation
-∂ᵢ(aⁱʲ ∂ⱼu) + bⁱ ∂ᵢu + c u ≤ f in Ω,
that is, a(u, v) ≤ ∫_Ω f v for every nonnegative v ∈ H¹₀(Ω), where a(·, ·) is the energy
form with drift b and mass coefficient c. Let a be measurable and uniformly elliptic with
constants 0 < λ ≤ Λ, and let b and c be measurable and bounded, with ‖b‖ ≤ β. For a level
k with w = (u - k)⁺ ∈ L²(Ω), every smooth ψ compactly supported in Ω satisfies
∫_Ω ψ² ‖∇w‖² ≤ 2 (2Λ/λ)² ∫_Ω ‖∇ψ‖² w² + (2β/λ)² ∫_Ω ψ² w² + (4/λ) ∫_Ω ψ² (f - c u) w.
Moving the drift and mass terms to the right-hand side makes u a subsolution of the principal
part with the effective forcing f - ⟪b, ∇u⟫ - c u, so the principal-part inequality applies.
The drift is then absorbed into the left-hand side by Young's inequality, at the cost of the
zeroth-order term (2β/λ)² ∫ ψ² w². The mass term is kept on the right-hand side next to the
forcing, where its sign is visible: under the sign condition c ≥ 0 and at a level k ≥ 0, the
product c u w is nonnegative, because w > 0 forces u > k ≥ 0, so the term can be dropped.
That is the form on concentric balls B(x₀, r) ⊆ B(x₀, R) ⊆ Ω for subsolutions of L u ≤ 0:
∫_{B(x₀, r)} ‖∇w‖² ≤ (2 (2Λ/λ)² (C/(R - r))² + (2β/λ)²) ∫_{B(x₀, R)} w²,
with C depending only on the dimension. Unlike the principal-part inequality
TauCeti.PDE.exists_setIntegral_ball_norm_gradient_posPartAbove_sq_le, this is not invariant
under dilations: the drift term is comparable to the cutoff term only on balls with βR/λ
bounded. This is the energy estimate that De Giorgi's iteration needs for equations with
lower-order terms. The boundedness hypothesis on c serves only to make the energy form
integrable.
Main declarations #
All four are in the namespace TauCeti.PDE.
UniformlyEllipticOn.setIntegral_sq_mul_norm_gradient_posPartAbove_sq_le_lowerOrder_of_memLp: Caccioppoli's inequality for(u - k)⁺at an arbitrary level with anL²hypothesis.UniformlyEllipticOn.setIntegral_sq_mul_norm_gradient_posPartAbove_sq_le_lowerOrder: the nonnegative-level specialization, whoseL²hypothesis is automatic.UniformlyEllipticOn.setIntegral_sq_mul_norm_gradient_posPartAbove_sq_le_lowerOrder_of_nonpos: the cutoff form for subsolutions ofL u ≤ 0withc ≥ 0, at a levelk ≥ 0, where the forcing and mass term drop out.exists_setIntegral_ball_norm_gradient_posPartAbove_sq_le_lowerOrder: the form on concentric balls for subsolutions ofL u ≤ 0withc ≥ 0.
References #
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, §8.5 and §8.6.
- Q. Han, F. Lin, Elliptic Partial Differential Equations, Chapter 4.
The Caccioppoli inequality for a positive truncation of a weak subsolution with
lower-order terms. Let a be measurable and uniformly elliptic on Ω with constants
0 < λ ≤ Λ, and let b and c be measurable with ‖b‖ ≤ β and ‖c‖ ≤ γ on Ω. Let
u ∈ H¹(Ω) satisfy
a(u, v) ≤ ∫_Ω f v
for every nonnegative v ∈ H¹₀(Ω), where a is the energy form of
-∂ᵢ(aⁱʲ ∂ⱼ ·) + bⁱ ∂ᵢ + c. At any level k for which w = (u - k)⁺ belongs to L²(Ω), every
smooth ψ compactly supported in Ω satisfies
∫_Ω ψ² ‖∇w‖² ≤ 2 (2Λ/λ)² ∫_Ω ‖∇ψ‖² w² + (2β/λ)² ∫_Ω ψ² w² + (4/λ) ∫_Ω ψ² (f - c u) w.
The bound γ on the mass coefficient does not enter the estimate.
The Caccioppoli inequality for a nonnegative-level truncation of a weak subsolution with
lower-order terms. This specializes
UniformlyEllipticOn.setIntegral_sq_mul_norm_gradient_posPartAbove_sq_le_lowerOrder_of_memLp;
the condition k ≥ 0 makes (u - k)⁺ ∈ L²(Ω) automatic even when Ω has infinite measure.
The zero-forcing Caccioppoli inequality for a nonnegative-level truncation with
lower-order terms. Let a be measurable and uniformly elliptic on Ω with constants
0 < λ ≤ Λ, let b and c be measurable with ‖b‖ ≤ β, ‖c‖ ≤ γ and c ≥ 0 on Ω, and let
u ∈ H¹(Ω) be a weak subsolution of -∂ᵢ(aⁱʲ ∂ⱼu) + bⁱ ∂ᵢu + c u ≤ 0. At any level k ≥ 0,
with w = (u - k)⁺, every smooth ψ compactly supported in Ω satisfies
∫_Ω ψ² ‖∇w‖² ≤ 2 (2Λ/λ)² ∫_Ω ‖∇ψ‖² w² + (2β/λ)² ∫_Ω ψ² w².
This specializes
UniformlyEllipticOn.setIntegral_sq_mul_norm_gradient_posPartAbove_sq_le_lowerOrder: the
mass term c u w is nonnegative because w > 0 forces u > k ≥ 0.
The Caccioppoli inequality on concentric balls, with lower-order terms. 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, let b and c be measurable with ‖b‖ ≤ β, ‖c‖ ≤ γ and c ≥ 0 on Ω, and let
u ∈ H¹(Ω) be a weak subsolution of -∂ᵢ(aⁱʲ ∂ⱼu) + bⁱ ∂ᵢu + c u ≤ 0. At any level k ≥ 0,
with w = (u - k)⁺, and for every pair of balls B(x₀, r) ⊆ B(x₀, R) ⊆ Ω with 0 < r < R,
∫_{B(x₀, r)} ‖∇w‖² ≤ (2 (2Λ/λ)² (C / (R - r))² + (2β/λ)²) ∫_{B(x₀, R)} w².