Local boundedness of weak subsolutions (De Giorgi) #
Let a be measurable and uniformly elliptic on Ω ⊆ ℝⁿ with constants 0 < λ ≤ Λ, and let
u ∈ H¹(Ω) be a weak subsolution of the divergence-form equation
-∂ⱼ(aⁱʲ ∂ᵢu) ≤ 0 in Ω,
meaning a(u, v) ≤ 0 for every nonnegative v ∈ H¹₀(Ω). This file proves De Giorgi's local
boundedness theorem: on every ball B(x₀, R) ⊆ Ω and above every level k,
u ≤ k + D R^{-n/2} ‖(u - k)⁺‖_{L²(B(x₀, R))} almost everywhere on B(x₀, R/2),
with D depending on λ, Λ, the dimension n ≥ 3 and the normalization of the additive
Haar measure used for the L² norm. No regularity of the coefficients beyond measurability
is used. This is the first half of the De Giorgi–Nash–Moser theorem; Hölder continuity is the
second.
The energy recursion setIntegral_sq_mul_max_sub_sq_le is useful when a Sobolev inequality
‖v‖_q ≤ S ‖∇v‖₂ is available on W^{1,2}_0(Ω) for some q > 2. It controls higher
truncation levels on smaller balls and yields the local bound below. The bound can be used as
the boundedness input for interior oscillation and Hölder regularity estimates.
The Sobolev inequality enters as a hypothesis in the general form, so the theorem applies to any
exponent q > 2 for which it is available. In dimension n ≥ 3 it is the
Gagliardo–Nirenberg–Sobolev inequality at q = 2n/(n - 2), which holds on every Ω with a
constant independent of Ω, but dependent on the normalization of the additive Haar measure.
Main declarations #
TauCeti.PDE.UniformlyEllipticOn.setIntegral_sq_mul_max_sub_sq_le: De Giorgi's energy recursion between two truncation levels.TauCeti.PDE.exists_ae_value_le_add_mul_rpow_mul_sqrt_setIntegral: local boundedness of weak subsolutions, under a Sobolev inequality with exponentq > 2.TauCeti.PDE.exists_ae_value_le_add_mul_rpow_mul_sqrt_setIntegral_of_inv_add_eq_inv: the scale-invariant boundu ≤ k + D R^{-n/2} ‖(u - k)⁺‖_{L²(B(x₀, R))}in dimensionn ≥ 3.TauCeti.PDE.exists_ae_abs_value_le_mul_rpow_mul_sqrt_setIntegral: the two-sided bound|u| ≤ D R^{-n/2} ‖u‖_{L²(B(x₀, R))}for weak solutions in dimensionn ≥ 3.
References #
- E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino (1957).
- Q. Han, F. Lin, Elliptic Partial Differential Equations, Chapter 4.
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Chapter 8.
De Giorgi's energy recursion. Let a be measurable and uniformly elliptic on Ω with
constants 0 < λ ≤ Λ, and suppose that W^{1,2}_0(Ω) satisfies a Sobolev inequality
‖v‖_q ≤ S ‖∇v‖₂ for some exponent q ≥ 2. Let u ∈ H¹(Ω) be a weak subsolution of
-∂ⱼ(aⁱʲ ∂ᵢu) ≤ 0, that is a(u, v) ≤ 0 for every nonnegative v ∈ H¹₀(Ω). Take levels
k < l with (u - k)⁺ ∈ L²(Ω), and a smooth ψ compactly supported in Ω with
‖∇ψ‖ ≤ G. Then, writing I = ∫_{Ω ∩ supp ψ} ((u - k)⁺)²,
∫_Ω ψ² ((u - l)⁺)² ≤ 2 (1 + (2Λ/λ)²) S² G² · I · (I / (l - k)²)^{1 - 2/q}.
The truncation at the higher level is controlled by a power 1 + (1 - 2/q) > 1 of the
truncation at the lower level: this superlinear gain is what drives De Giorgi's iteration.
Local boundedness of weak subsolutions (De Giorgi). Fix ellipticity constants λ, Λ, an
exponent q > 2 and a constant S. There is D > 0, depending only on these (and the
dimension), such that the following holds. Let a be measurable and uniformly elliptic on Ω
with constants λ, Λ, suppose that ‖v‖_q ≤ S ‖∇v‖₂ for every v ∈ W^{1,2}_0(Ω), and let
u ∈ H¹(Ω) be a weak subsolution of -∂ⱼ(aⁱʲ ∂ᵢu) ≤ 0, that is a(u, v) ≤ 0 for every
nonnegative v ∈ H¹₀(Ω). Then for every level k and every ball B(x₀, R) ⊆ Ω,
u ≤ k + D R^{-1/α} (∫_{B(x₀, R)} ((u - k)⁺)²)^{1/2} almost everywhere on B(x₀, R/2),
where α = 1 - 2/q. For n ≥ 3 and the Sobolev exponent q = 2n/(n - 2),
α = 2/n and the bound is the classical u ≤ k + D R^{-n/2} ‖(u - k)⁺‖_{L²(B(x₀, R))}; see
TauCeti.PDE.exists_ae_value_le_add_mul_rpow_mul_sqrt_setIntegral_of_inv_add_eq_inv.
No regularity of the coefficients beyond measurability, and no boundary condition on u, is
assumed.
Local boundedness of weak subsolutions in dimension n ≥ 3 (De Giorgi). Let 2* be the
Sobolev exponent of W^{1,2} in dimension n, so that 1/2* + 1/n = 1/2 and 2* < ∞ (this
forces n ≥ 3). There is D > 0, depending on λ, Λ, the dimension and the normalization
of the additive Haar measure mu, such that for every measurable, uniformly elliptic a on
Ω with constants λ, Λ, every weak subsolution u ∈ H¹(Ω) of
-∂ⱼ(aⁱʲ ∂ᵢu) ≤ 0, every level k and every ball B(x₀, R) ⊆ Ω,
u ≤ k + D R^{-n/2} (∫_{B(x₀, R)} ((u - k)⁺)²)^{1/2} almost everywhere on B(x₀, R/2).
The Sobolev inequality needed by
TauCeti.PDE.exists_ae_value_le_add_mul_rpow_mul_sqrt_setIntegral is
the Gagliardo–Nirenberg–Sobolev inequality on W^{1,2}_0(Ω), whose constant does not depend on
Ω, but does depend on mu; this makes D independent of the domain.
Local boundedness of weak solutions in dimension n ≥ 3 (De Giorgi). Let 2* be the
Sobolev exponent of W^{1,2} in dimension n, so that 1/2* + 1/n = 1/2 and 2* < ∞ (this
forces n ≥ 3). There is D > 0, depending on λ, Λ, the dimension and the normalization
of the additive Haar measure mu, such that for every measurable, uniformly elliptic a on
Ω with constants λ, Λ, every weak solution u ∈ H¹(Ω) of -∂ⱼ(aⁱʲ ∂ᵢu) = 0, that is
a(u, v) = 0 for every v ∈ H¹₀(Ω), and every ball B(x₀, R) ⊆ Ω,
|u| ≤ D R^{-n/2} ‖u‖_{L²(B(x₀, R))} almost everywhere on B(x₀, R/2).
This is the two-sided form of
TauCeti.PDE.exists_ae_value_le_add_mul_rpow_mul_sqrt_setIntegral_of_inv_add_eq_inv, obtained
by applying it at the level 0 to the weak subsolutions u and -u.