The Caccioppoli inequality for truncations of weak subsolutions #
Let u ∈ H¹(Ω) be a weak subsolution of the divergence-form equation
-∂ⱼ(aⁱʲ ∂ᵢu) ≤ f in Ω.
Thus the weak energy inequality holds against every nonnegative test function in H¹₀(Ω).
For a level k, put w = (u - k)⁺ and assume w ∈ L²(Ω). If a is measurable and uniformly
elliptic with constants 0 < λ ≤ Λ, then every smooth cutoff ψ compactly supported in Ω
satisfies
∫_Ω ψ² ‖∇w‖² ≤ (2Λ/λ)² ∫_Ω ‖∇ψ‖² w² + (2/λ) ∫_Ω ψ² f w.
The test function is ψ²w. It is nonnegative and belongs to H¹₀(Ω) because the cutoff is
compactly supported. On {u > k}, the weak gradient of w equals that of u; off this set,
both the value and weak gradient of w vanish. This reduces the pointwise energy estimate to
the same absorption inequality as for weak solutions.
This truncation estimate is the energy input to De Giorgi iteration and to the corresponding weak maximum-principle argument.
Main declarations #
TauCeti.PDE.UniformlyEllipticOn.setIntegral_sq_mul_norm_gradient_posPartAbove_sq_le_of_memLp: Caccioppoli's inequality for(u - k)⁺at an arbitrary level with anL²hypothesis.TauCeti.PDE.UniformlyEllipticOn.setIntegral_sq_mul_norm_gradient_posPartAbove_sq_le: the nonnegative-level specialization, whoseL²hypothesis is automatic.TauCeti.PDE.exists_setIntegral_ball_norm_gradient_posPartAbove_sq_le: the form on concentric ballsB(x₀, r) ⊆ B(x₀, R),∫_{B(x₀, r)} ‖∇w‖² ≤ (2Λ/λ)² (c/(R - r))² ∫_{B(x₀, R)} w².
References #
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, §8.6.
The Caccioppoli inequality for a positive truncation of a weak subsolution.
Let a be measurable and uniformly elliptic on Ω with constants 0 < λ ≤ Λ, and let
u ∈ H¹(Ω) satisfy
a(u, v) ≤ ∫_Ω f v
for every nonnegative v ∈ H¹₀(Ω). 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 ≤ (2Λ/λ)² ∫_Ω ‖∇ψ‖² w² + (2/λ) ∫_Ω ψ² f w.
No boundary regularity or coefficient regularity beyond measurability is assumed.
The zero-forcing Caccioppoli inequality for a positive truncation of a weak
subsolution. This is the specialization of
UniformlyEllipticOn.setIntegral_sq_mul_norm_gradient_posPartAbove_sq_le_of_memLp to
-∂ⱼ(aⁱʲ ∂ᵢu) ≤ 0.
The Caccioppoli inequality for a nonnegative-level truncation of a weak subsolution.
This specializes
UniformlyEllipticOn.setIntegral_sq_mul_norm_gradient_posPartAbove_sq_le_of_memLp; the
condition k ≥ 0 makes (u - k)⁺ ∈ L²(Ω) automatic even when Ω has infinite measure.
The 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 subsolution of -∂ⱼ(aⁱʲ ∂ᵢu) ≤ 0. At any level k with
w = (u - k)⁺ ∈ L²(Ω), and for every pair of balls B(x₀, r) ⊆ B(x₀, R) ⊆ Ω with
0 < r < R,
∫_{B(x₀, r)} ‖∇w‖² ≤ (2Λ/λ)² (c / (R - r))² ∫_{B(x₀, R)} w².