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
-∂ᵢ(aⁱʲ ∂ⱼu) ≤ 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 ≥ 1 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 _of_local_subsolution and _of_local_solution forms allow an ambient Sobolev function
while requiring the coefficient hypotheses and the weak equation only on the selected ball,
with no assumptions on the coefficients or equation outside that ball.
The energy recursion setIntegral_sq_mul_max_sub_sq_le_of_caccioppoli is useful when a Sobolev
inequality ‖v‖_q ≤ S ‖∇v‖₂ is available on W^{1,2}_0(B(x₀, R)) 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.
De Giorgi's classes #
The equation enters De Giorgi's iteration only through the Caccioppoli inequality for the
truncations w = (u - l)⁺ above the level k,
∫ ψ² ‖∇w‖² ≤ A ∫ ‖∇ψ‖² w² + B ∫ ψ² w² for every smooth cutoff ψ,
so the iteration is carried out for every u with this property (TauCeti.PDE.CaccioppoliAbove),
with constants A, B ≥ 0. Weak subsolutions of -∂ᵢ(aⁱʲ ∂ⱼu) ≤ 0 have it at every level with
A = (2Λ/λ)² and B = 0. The zeroth-order term B ∫ ψ² w² is what a drift contributes, and it
costs a factor (1 + A + B R²)ⁿ in the bound on B(x₀, R/2).
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, with a constant D S^{1/α}, α = 1 - 2/q. For the
scale-invariant bound it is applied on the ball B(x₀, R) itself, where the Sobolev inequality
for sets of finite measure,
TauCeti.W1p.eLpNorm_value_le_mul_rpow_mul_enorm_gradient_of_eq_ball, holds for some q > 2
in every dimension, with S a multiple of R ^ (1 - n (1/2 - 1/q)). Then S^{1/α} R^{-1/α} is a
multiple of R^{-n/2}. In dimension n ≥ 3 the Gagliardo–Nirenberg–Sobolev inequality would
serve as well, but in dimensions 1 and 2 there is no critical exponent and the dependence of
S on the ball is essential.
Lower exponents #
The L² norm on the right can be replaced by any Lᵖ norm with 0 < p ≤ 2, at the cost of
the scaling exponent -n/2 becoming -n/p:
u ≤ k + D R^{-n/p} ‖(u - k)⁺‖_{Lᵖ(B(x₀, R))} almost everywhere on B(x₀, R/2).
Writing F(r) for the essential supremum of (u - k)⁺ on B(x₀, r), the L² bound on
concentric balls r < ρ and the interpolation ‖w‖²_{L²} ≤ F(ρ)^{2-p} ‖w‖ᵖ_{Lᵖ} give
F(r) ≤ F(ρ)/2 + C (ρ - r)^{-n/p} ‖(u - k)⁺‖_{Lᵖ}, and the absorption lemma
TauCeti.exists_le_mul_mul_rpow_add_of_le_mul_add removes the term F(ρ)/2. Small p is
what Moser's proof of the Harnack inequality needs, where the supremum bound for subsolutions is
matched with an infimum bound for supersolutions through a small positive power.
Main declarations #
TauCeti.PDE.setIntegral_sq_mul_max_sub_sq_le_of_caccioppoli: De Giorgi's energy recursion between two truncation levels, for a function satisfying the Caccioppoli inequality.TauCeti.PDE.CaccioppoliAbove: De Giorgi's class above a level, the truncations satisfying the Caccioppoli inequality.TauCeti.PDE.exists_ae_value_le_add_mul_pow_mul_rpow_mul_sqrt_setIntegral_of_caccioppoli: the scale-invariant local bound for functions in De Giorgi's classes, in every dimensionn ≥ 1.TauCeti.PDE.ae_abs_value_le_mul_sqrt_setIntegral_of_ae_value_le: one-sided bounds foruand-ugive a two-sided bound.TauCeti.PDE.UniformlyEllipticOn.setIntegral_sq_mul_max_sub_sq_le: the energy recursion for weak subsolutions.TauCeti.PDE.UniformlyEllipticOn.caccioppoliAbove: weak subsolutions belong to De Giorgi's class above every level.TauCeti.PDE.exists_ae_value_le_add_mul_rpow_mul_rpow_mul_sqrt_setIntegral_of_forall_eLpNorm_le: local boundedness of weak subsolutions on a ball, under a Sobolev inequality with exponentq > 2and constantSon that ball, with a constant homogeneous inS.TauCeti.PDE.exists_ae_value_le_add_mul_rpow_mul_sqrt_setIntegral: the scale-invariant boundu ≤ k + D R^{-n/2} ‖(u - k)⁺‖_{L²(B(x₀, R))}in every dimensionn ≥ 1.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.TauCeti.PDE.exists_ae_le_mul_rpow_mul_setIntegral_rpow_of_forall_ball: localL²bounds for the supremum on half balls imply the same bounds with anyLᵖnorm,0 < p ≤ 2.TauCeti.PDE.exists_ae_value_le_add_mul_rpow_mul_setIntegral_rpow_of_forall_eLpNorm_le: local boundedness of weak subsolutions with theLᵖnorm of(u - k)⁺,0 < p ≤ 2, on the right, under a Sobolev inequality on every ball insideΩ.TauCeti.PDE.exists_ae_value_le_add_mul_rpow_mul_setIntegral_rpow: the boundu ≤ k + D R^{-n/p} ‖(u - k)⁺‖_{Lᵖ(B(x₀, R))}in every dimensionn ≥ 1.
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 classes and iteration in L² #
De Giorgi's energy recursion. 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), take levels k < l with
(u - k)⁺ ∈ L²(Ω), and a smooth ψ compactly supported in Ω with ‖∇ψ‖ ≤ G. Write
w = (u - l)⁺ and I = ∫_{Ω ∩ supp ψ} ((u - k)⁺)², and suppose that w satisfies the
Caccioppoli inequality
∫_Ω ψ² ‖∇w‖² ≤ A ∫_Ω ‖∇ψ‖² w² + B I
with A ≥ 0. Then
∫_Ω ψ² w² ≤ 2 ((1 + A) G² + B) S² · 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. For
-∂ᵢ(aⁱʲ ∂ⱼu) ≤ 0 the Caccioppoli inequality holds with A = (2Λ/λ)² and B = 0
(TauCeti.PDE.UniformlyEllipticOn.setIntegral_sq_mul_max_sub_sq_le). For equations with a
drift it carries a zeroth-order term B ∫_Ω ψ² w², which is at most B I when |ψ| ≤ 1.
De Giorgi's class above a level. CaccioppoliAbove u k A B states that the truncations
w = (u - l)⁺ of u ∈ H¹(Ω) at every level l ≥ k with w ∈ L²(Ω) satisfy the Caccioppoli
inequality
∫_Ω ψ² ‖∇w‖² ≤ A ∫_Ω ‖∇ψ‖² w² + B ∫_Ω ψ² w²
for every smooth ψ compactly supported in Ω. This is the only way the equation enters
De Giorgi's iteration.
Instances For
Membership in De Giorgi's class above k: the Caccioppoli inequality for the truncations
(u - l)⁺ at every level l ≥ k with (u - l)⁺ ∈ L²(Ω), against every admissible cutoff.
For u in De Giorgi's class above k, the truncation w = (u - l)⁺ at a level l ≥ k
with w ∈ L²(Ω) satisfies ∫_Ω ψ² ‖∇w‖² ≤ A ∫_Ω ‖∇ψ‖² w² + B ∫_Ω ψ² w² for every smooth ψ
compactly supported in Ω.
De Giorgi's class shrinks as the level rises and grows with the constants A and B.
Local boundedness from the Caccioppoli inequality (De Giorgi). In every dimension
n ≥ 1 there is D > 0, depending on the dimension and the normalization of the additive Haar
measure mu, such that the following holds. Let u ∈ H¹(Ω), let B(x₀, R) ⊆ Ω and k be a
level, and suppose that the restriction of u to the ball satisfies the Caccioppoli inequality
at every level l ≥ k: with w = (u - l)⁺, every smooth ψ compactly supported in B(x₀, R)
satisfies
∫ ψ² ‖∇w‖² ≤ A ∫ ‖∇ψ‖² w² + B ∫ ψ² w²
over B(x₀, R), with A, B ≥ 0. Then
u ≤ k + D (1 + A + B R²)ⁿ R^{-n/2} (∫_{B(x₀, R)} ((u - k)⁺)²)^{1/2} almost everywhere on
B(x₀, R/2).
Functions satisfying such Caccioppoli inequalities form De Giorgi's classes
(TauCeti.PDE.CaccioppoliAbove). Weak subsolutions of -∂ᵢ(aⁱʲ ∂ⱼu) ≤ 0 belong to them at every
level with B = 0, which gives the scale-invariant bound
TauCeti.PDE.exists_ae_value_le_add_mul_rpow_mul_sqrt_setIntegral. A drift contributes the
zeroth-order term B ∫ ψ² w², which is why the bound depends on B R².
The Sobolev inequality that drives De Giorgi's iteration is the one on W^{1,2}_0(B(x₀, R)),
TauCeti.W1p.eLpNorm_value_le_mul_rpow_mul_enorm_gradient_of_eq_ball, at an exponent q > 2
with 1/2 - 1/q < 1/n, which exists in every dimension. Its constant is a multiple of
R ^ (1 - n (1/2 - 1/q)), and this power of R is exactly what turns the bound into the
scale-invariant one above.
From one-sided to two-sided local bounds. If u ∈ H¹(Ω) and -u both satisfy the
one-sided local bound above the level 0 on B(x₀, R) ⊆ Ω with the same constant K ≥ 0, that
is ±u ≤ K ‖(±u)⁺‖_{L²(B(x₀, R))} almost everywhere on B(x₀, R/2), then
|u| ≤ K ‖u‖_{L²(B(x₀, R))} almost everywhere on B(x₀, R/2).
This is how local boundedness of weak subsolutions gives local boundedness of weak solutions of
linear equations, for which u and -u are both subsolutions.
Local boundedness of weak subsolutions and solutions in L² #
De Giorgi's energy recursion for weak subsolutions. 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
-∂ᵢ(aⁱʲ ∂ⱼu) ≤ 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}.
This is TauCeti.PDE.setIntegral_sq_mul_max_sub_sq_le_of_caccioppoli with the Caccioppoli
inequality for truncations of subsolutions.
Weak subsolutions of -∂ᵢ(aⁱʲ ∂ⱼu) ≤ 0 belong to De Giorgi's class above every level, with
A = (2Λ/λ)² and B = 0.
Local boundedness of weak subsolutions under a Sobolev inequality (De Giorgi). Fix
ellipticity constants λ, Λ and an exponent q > 2, and let α = 1 - 2/q. 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, and let u ∈ H¹(Ω) be a weak
subsolution of -∂ᵢ(aⁱʲ ∂ⱼu) ≤ 0, that is a(u, v) ≤ 0 for every nonnegative v ∈ H¹₀(Ω).
If B(x₀, R) ⊆ Ω and ‖v‖_q ≤ S ‖∇v‖₂ for every v ∈ W^{1,2}_0(B(x₀, R)), then for every
level k,
u ≤ k + D S^{1/α} R^{-1/α} (∫_{B(x₀, R)} ((u - k)⁺)²)^{1/2} almost everywhere on B(x₀, R/2).
The Sobolev inequality is only needed on the ball itself, so S may depend on the ball, and D
does not depend on S. This is what makes the bound scale-invariant: on B(x₀, R) the Sobolev
inequality for sets of finite measure holds with S a multiple of R ^ (1 - n (1/2 - 1/q)),
and S^{1/α} R^{-1/α} is then a multiple of R^{-n/2}; in every dimension that bound is
TauCeti.PDE.exists_ae_value_le_add_mul_rpow_mul_sqrt_setIntegral.
No regularity of the coefficients beyond measurability, and no boundary condition on u, is
assumed. Restricting u to the ball, which has finite measure, makes (u - k)⁺ square
integrable.
Local boundedness of weak subsolutions (De Giorgi). In every dimension n ≥ 1 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 -∂ᵢ(aⁱʲ ∂ⱼu) ≤ 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).
This is TauCeti.PDE.exists_ae_value_le_add_mul_pow_mul_rpow_mul_sqrt_setIntegral_of_caccioppoli
with the Caccioppoli inequality for truncations of subsolutions, which holds at every level with
A = (2Λ/λ)² and B = 0.
Local boundedness of weak solutions (De Giorgi). In every dimension n ≥ 1 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 -∂ᵢ(aⁱʲ ∂ⱼu) = 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,
obtained by applying it at the level 0 to the weak subsolutions u and -u.
Local boundedness in terms of Lᵖ norms #
Lᵖ local bounds from L² local bounds. Fix e, D and 0 < p ≤ 2. There is
C > 0, depending only on these, such that the following holds. Let w ≥ 0 be square
integrable on B(x₀, R) and suppose that on every ball B(y, s) ⊆ B(x₀, R),
w ≤ D s^e ‖w‖_{L²(B(y, s))} almost everywhere on B(y, s/2).
Then w ≤ C R^{2e/p} ‖w‖_{Lᵖ(B(x₀, R))} almost everywhere on B(x₀, R/2).
This is how local boundedness of subsolutions, proved by De Giorgi's or Moser's iteration with
the L² norm on the right, is upgraded to every Lᵖ norm with 0 < p < 2: the scaling
exponent e = -n/2 of the L² bound becomes -n/p.
Local boundedness of weak subsolutions in Lᵖ under a Sobolev inequality (De Giorgi).
Fix ellipticity constants λ, Λ, an exponent q > 2, a constant S and 0 < p ≤ 2. 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(B) and every ball B ⊆ Ω, and let u ∈ H¹(Ω) be a weak subsolution
of -∂ᵢ(aⁱʲ ∂ⱼu) ≤ 0. Then for every level k and every ball B(x₀, R) ⊆ Ω,
u ≤ k + D R^{-2/(α p)} (∫_{B(x₀, R)} ((u - k)⁺)ᵖ)^{1/p} almost everywhere on B(x₀, R/2),
where α = 1 - 2/q. It follows from
TauCeti.PDE.exists_ae_value_le_add_mul_rpow_mul_rpow_mul_sqrt_setIntegral_of_forall_eLpNorm_le
on every ball inside Ω, which is why the Sobolev inequality is assumed on all of them.
Local boundedness of weak subsolutions in Lᵖ (De Giorgi). Let 0 < p ≤ 2. In every
dimension n ≥ 1 there is D > 0, depending on λ, Λ, p, 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
-∂ᵢ(aⁱʲ ∂ⱼu) ≤ 0, every level k and every ball B(x₀, R) ⊆ Ω,
u ≤ k + D R^{-n/p} (∫_{B(x₀, R)} ((u - k)⁺)ᵖ)^{1/p} almost everywhere on B(x₀, R/2).
At p = 2 this is TauCeti.PDE.exists_ae_value_le_add_mul_rpow_mul_sqrt_setIntegral; small
p is the form in which local boundedness enters Moser's Harnack inequality.
Local boundedness with hypotheses only on the ball. For every 0 < p ≤ 2 and every
positive dimension,
the coefficients need be elliptic and measurable only on B(x₀, R), and the restriction of
u to that ball need be a weak subsolution only there. The bound applies to the original
ambient-domain representative of u; its constant is independent of the radius.
The two-sided local bound with hypotheses only on the ball. A weak solution after
restriction to B(x₀, R) satisfies the scale-invariant absolute-value estimate on its half
ball, without ellipticity, measurability or the equation being required outside the ball.