Global boundedness of weak subsolutions (Stampacchia) #
Let Ω ⊆ ℝⁿ be open with finite measure, let a be measurable and uniformly elliptic on Ω
with constants 0 < λ ≤ Λ, and let c be bounded, measurable and nonnegative. This file proves
the weak maximum principle with a right-hand side: if u ∈ H¹(Ω) is a weak subsolution of
-∂ᵢ(aⁱʲ ∂ⱼu) + c u ≤ f in Ω, with f ∈ Lʳ(Ω), r > 1 and r > n/2,
meaning a(u, v) ≤ ∫_Ω f v for every nonnegative v ∈ H¹₀(Ω), and u ≤ k on ∂Ω for a level
k ≥ 0, in the sense that (u - k)⁺ ∈ H¹₀(Ω), then
u ≤ k + C ‖f‖_{Lʳ(Ω)} / λ · |Ω| ^ (2/n - 1/r) almost everywhere in Ω,
with C depending only on n, r and the normalization of the Haar measure. In particular the
weak solution u ∈ H¹₀(Ω) of the Dirichlet problem -∂ᵢ(aⁱʲ ∂ⱼu) + c u = f with
f ∈ Lʳ(Ω), meaning a(u, v) = ∫_Ω f v for every v ∈ H¹₀(Ω), is bounded, with
|u| ≤ C ‖f‖_{Lʳ} / λ · |Ω| ^ (2/n - 1/r). No regularity of ∂Ω and no regularity of the
coefficients beyond measurability is used, and the power of |Ω| makes the estimate invariant
under dilations.
This is the bound that lets De Giorgi's interior estimates absorb a right-hand side: on a ball
B ⊆ Ω, a solution of -∂ᵢ(aⁱʲ ∂ⱼu) = f differs from a solution of the homogeneous equation by
the solution of the Dirichlet problem on B with data f, and the bound above controls that
difference by R ^ (2 - n/r) ‖f‖_{Lʳ}.
Stampacchia's method #
For a level l ≥ k, the truncation w = (u - l)⁺ lies in H¹₀(Ω) and is a nonnegative test
function, and its gradient is ∇u on {u > l} and zero elsewhere. Testing the subsolution
inequality against it, ellipticity and c ≥ 0 give λ ‖∇w‖² ≤ ∫ f w. A Sobolev inequality
‖w‖_{Lˢ} ≤ S ‖∇w‖_{L²} on H¹₀(Ω) and Hölder's inequality on the superlevel set {u > l},
off which w vanishes, turn this into
‖(u - l)⁺‖_{Lˢ} ≤ S² ‖f‖_{Lʳ} / λ · |{u > l}| ^ (1 - 1/r - 1/s).
Chebyshev's inequality on {u > h}, where (u - l)⁺ ≥ h - l, then gives, for φ l = |{u > l}|
and M = S² ‖f‖_{Lʳ} / λ,
φ h ≤ (M / (h - l)) ^ s φ l ^ β for k ≤ l < h, where β = s (1 - 1/r - 1/s).
When s (1 - 1/r) > 2, the exponent β exceeds 1, and Stampacchia's lemma
(TauCeti.apply_add_eq_zero_of_le_div_sub_rpow_mul_rpow) shows that φ vanishes at a finite
level above k. In every dimension the Sobolev inequality on sets of finite measure
(TauCeti.W1p.eLpNorm_value_le_mul_measure_rpow_mul_enorm_gradient_of_lt) provides such an
exponent s whenever r > 1 and r > n/2, with S a multiple of |Ω| ^ (1/n - 1/2 + 1/s).
Only the energy estimate for ‖(u - l)⁺‖_{Lˢ} depends on the right-hand side; the passage from it
to the bound on u is shared by both kinds of data below.
A right-hand side in divergence form #
The same holds for weak subsolutions of -∂ᵢ(aⁱʲ ∂ⱼu) + c u ≤ ∂ᵢFⁱ with F ∈ L^q(Ω; ℝⁿ),
q > n and q > 2, meaning a(u, v) ≤ -∫_Ω F · ∇v for every nonnegative v ∈ H¹₀(Ω):
u ≤ k + C ‖F‖_{L^q(Ω)} / λ · |Ω| ^ (1/n - 1/q) almost everywhere in Ω.
Testing against w = (u - l)⁺ now gives λ ‖∇w‖² ≤ ∫ ‖F‖ ‖∇w‖, and since ∇w vanishes off
{u > l}, Hölder's inequality bounds the right side by
‖F‖_{L^q} ‖∇w‖_{L²} |{u > l}| ^ (1/2 - 1/q). After cancelling one factor of ‖∇w‖_{L²}, the
Sobolev inequality gives ‖(u - l)⁺‖_{Lˢ} ≤ S ‖F‖_{L^q} / λ · |{u > l}| ^ (1/2 - 1/q), and
Stampacchia's lemma applies once s (1/2 - 1/q) > 1. On a ball B of radius ρ this bounds the
solution of the Dirichlet problem on B with datum ∂ᵢFⁱ by a multiple of
ρ ^ (1 - n/q) ‖F‖_{L^q}. That is the divergence-form counterpart of the bound above, needed when
De Giorgi's interior estimates absorb a divergence-form right-hand side.
Main declarations #
TauCeti.PDE.UniformlyEllipticOn.ae_value_le_add_of_forall_eLpNorm_le: the bound with an explicit constant, under a Sobolev inequality onH¹₀(Ω)with exponentsand constantS.TauCeti.PDE.exists_ae_value_le_add_mul_div_mul_rpow: the scale-invariant bound in every dimension, forr > max 1 (n/2).TauCeti.PDE.exists_ae_abs_value_le_mul_div_mul_rpow: the two-sided bound for weak solutions of the Dirichlet problem.TauCeti.PDE.UniformlyEllipticOn.ae_value_le_add_of_forall_eLpNorm_le_divergence: the bound with an explicit constant for a right-hand side∂ᵢFⁱin divergence form,F ∈ L^qwithq ≥ 2, under a Sobolev inequality onH¹₀(Ω)with exponentsand constantS, wheres (1/2 - 1/q) > 1.TauCeti.PDE.exists_ae_value_le_add_mul_div_mul_rpow_divergence: the scale-invariant bound for a right-hand side in divergence form in every dimension, forq > max 2 n.TauCeti.PDE.exists_ae_abs_value_le_mul_div_mul_rpow_divergence: the two-sided bound for weak solutions of the Dirichlet problem with a right-hand side in divergence form,q > max 2 n.
References #
- G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier 15 (1965).
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Theorem 8.15 and Theorem 8.16.
- D. Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Chapter II, Appendix B.
Stampacchia's global bound. Let Ω have finite measure, let a be measurable and
uniformly elliptic on Ω with constants 0 < λ ≤ Λ, and let c be bounded, measurable and
nonnegative. Suppose that W^{1,2}_0(Ω) satisfies a Sobolev inequality ‖v‖_{Lˢ} ≤ S ‖∇v‖_{L²},
and let f ∈ Lʳ(Ω) with s (1 - 1/r) > 2. If u ∈ H¹(Ω) is a weak subsolution of
-∂ᵢ(aⁱʲ ∂ⱼu) + c u ≤ f in Ω,
that is a(u, v) ≤ ∫_Ω f v for every nonnegative v ∈ H¹₀(Ω), and u ≤ k on ∂Ω for a level
k ≥ 0, in the sense that (u - k)⁺ ∈ H¹₀(Ω), then almost everywhere in Ω
u ≤ k + 2 ^ (β / (β - 1)) S² ‖f‖_{Lʳ} / λ · |Ω| ^ (1 - 1/r - 2/s), where β = s (1 - 1/r) - 1.
No regularity of ∂Ω or of the coefficients beyond measurability is needed.
Stampacchia's global bound, in every dimension. Let n ≥ 1 be the dimension and let
r > 1 with r > n/2. There is C > 0, depending only on n, r and the normalization of the
additive Haar measure mu, with the following property. Let Ω have finite measure, let a be
measurable and uniformly elliptic on Ω with constants 0 < λ ≤ Λ, let c be bounded,
measurable and nonnegative, and let f ∈ Lʳ(Ω). If u ∈ H¹(Ω) is a weak subsolution of
-∂ᵢ(aⁱʲ ∂ⱼu) + c u ≤ f in Ω
with u ≤ k on ∂Ω for a level k ≥ 0, in the sense that (u - k)⁺ ∈ H¹₀(Ω), then almost
everywhere in Ω
u ≤ k + C ‖f‖_{Lʳ(Ω)} / λ · |Ω| ^ (2/n - 1/r).
The bound is invariant under dilations of Ω. The Sobolev inequality behind it is the one on
sets of finite measure, TauCeti.W1p.eLpNorm_value_le_mul_measure_rpow_mul_enorm_gradient_of_lt,
which holds in every dimension.
Global boundedness of weak Dirichlet solutions, in every dimension. Let n ≥ 1 be the
dimension and let r > 1 with r > n/2. There is C > 0, depending only on n, r and the
normalization of mu, such that on every Ω of finite measure, for measurable, uniformly
elliptic a with constants 0 < λ ≤ Λ and bounded, measurable, nonnegative c, the weak
solution u ∈ H¹₀(Ω) of the Dirichlet problem -∂ᵢ(aⁱʲ ∂ⱼu) + c u = f with f ∈ Lʳ(Ω),
that is a(u, v) = ∫_Ω f v for every v ∈ H¹₀(Ω), satisfies, almost everywhere in Ω,
|u| ≤ C ‖f‖_{Lʳ(Ω)} / λ · |Ω| ^ (2/n - 1/r).
No integrability of f beyond Lʳ is assumed; for f ∈ L²(Ω) the hypothesis on u is
TauCeti.PDE.IsWeakSolutionDirichlet a 0 c f u, by TauCeti.PDE.isWeakSolutionDirichlet_iff.
This is the two-sided form of TauCeti.PDE.exists_ae_value_le_add_mul_div_mul_rpow at the level
k = 0, applied to u and to -u.
A right-hand side in divergence form #
Stampacchia's global bound for a right-hand side in divergence form. Let Ω have finite
measure, let a be measurable and uniformly elliptic on Ω with constants 0 < λ ≤ Λ, and let
c be bounded, measurable and nonnegative. Suppose that W^{1,2}_0(Ω) satisfies a Sobolev
inequality ‖v‖_{Lˢ} ≤ S ‖∇v‖_{L²}, and let F ∈ L^q(Ω; ℝⁿ) with q ≥ 2 and
s (1/2 - 1/q) > 1. If u ∈ H¹(Ω) is a weak subsolution of
-∂ᵢ(aⁱʲ ∂ⱼu) + c u ≤ ∂ᵢFⁱ in Ω,
that is a(u, v) ≤ -∫_Ω F · ∇v for every nonnegative v ∈ H¹₀(Ω), and u ≤ k on ∂Ω for a
level k ≥ 0, in the sense that (u - k)⁺ ∈ H¹₀(Ω), then almost everywhere in Ω
u ≤ k + 2 ^ (β / (β - 1)) S ‖F‖_{L^q} / λ · |Ω| ^ (1/2 - 1/q - 1/s), where
β = s (1/2 - 1/q).
No regularity of ∂Ω or of the coefficients beyond measurability is needed.
Stampacchia's global bound for a right-hand side in divergence form, in every dimension.
Let n ≥ 1 be the dimension and let q > n with q > 2. There is C > 0, depending only on
n, q and the normalization of the additive Haar measure mu, with the following property.
Let Ω have finite measure, let a be measurable and uniformly elliptic on Ω with constants
0 < λ ≤ Λ, let c be bounded, measurable and nonnegative, and let F ∈ L^q(Ω; ℝⁿ). If
u ∈ H¹(Ω) is a weak subsolution of
-∂ᵢ(aⁱʲ ∂ⱼu) + c u ≤ ∂ᵢFⁱ in Ω
with u ≤ k on ∂Ω for a level k ≥ 0, in the sense that (u - k)⁺ ∈ H¹₀(Ω), then almost
everywhere in Ω
u ≤ k + C ‖F‖_{L^q(Ω)} / λ · |Ω| ^ (1/n - 1/q).
The bound is invariant under dilations of Ω; on a ball of radius ρ the factor
|Ω| ^ (1/n - 1/q) is a multiple of the Morrey gain ρ ^ (1 - n/q). The condition q > 2
matters only for n = 1; it comes from the Sobolev exponents s = (1/2 - δ)⁻¹, δ < 1/2, which
the method needs to satisfy s (1/2 - 1/q) > 1.
Global boundedness of weak Dirichlet solutions with a right-hand side in divergence form,
in every dimension. Let n ≥ 1 be the dimension and let q > n with q > 2. There is
C > 0, depending only on n, q and the normalization of mu, such that on every Ω of
finite measure, for measurable, uniformly elliptic a with constants 0 < λ ≤ Λ and bounded,
measurable, nonnegative c, the weak solution u ∈ H¹₀(Ω) of the Dirichlet problem
-∂ᵢ(aⁱʲ ∂ⱼu) + c u = ∂ᵢFⁱ with F ∈ L^q(Ω; ℝⁿ), that is a(u, v) = -∫_Ω F · ∇v for every
v ∈ H¹₀(Ω), satisfies, almost everywhere in Ω,
|u| ≤ C ‖F‖_{L^q(Ω)} / λ · |Ω| ^ (1/n - 1/q).
This is the two-sided form of TauCeti.PDE.exists_ae_value_le_add_mul_div_mul_rpow_divergence at
the level k = 0, applied to u and to -u.