Hölder continuity of weak solutions with a right-hand side #
Let a be measurable and uniformly elliptic on an open set Ω ⊆ ℝⁿ, n ≥ 1, with constants
0 < λ ≤ Λ, and let f ∈ Lʳ with r > 1 and r > n/2. This file extends De Giorgi's theorem
TauCeti.PDE.exists_holderOnWith_preciseRepresentative from -∂ᵢ(aⁱʲ ∂ⱼu) = 0 to weak solutions
u ∈ H¹(Ω) of
-∂ᵢ(aⁱʲ ∂ⱼu) = f in Ω, that is a(u, v) = ∫_Ω f v for every v ∈ H¹₀(Ω).
Such a solution is locally bounded and has a locally Hölder continuous representative, with an
exponent depending only on λ, Λ, n, r and the normalization of the Haar measure, and with
constants controlled by ‖u‖_{L²} and ‖f‖_{Lʳ}/λ. No regularity of the coefficients beyond
measurability is assumed.
The right-hand side is split off one ball at a time. On B = B(x₀, ρ) ⊆ Ω, the Dirichlet problem
-∂ᵢ(aⁱʲ ∂ⱼw) = f in B, w ∈ H¹₀(B), has a solution
(TauCeti.PDE.UniformlyEllipticOn.existsUnique_forall_energyFormH1_eq_setIntegral_of_memLp), and
Stampacchia's bound TauCeti.PDE.exists_ae_abs_value_le_mul_div_mul_rpow gives
|w| ≤ C ρ^(2 - n/r) ‖f‖_{Lʳ(B)} / λ. The difference h = u - w solves the homogeneous equation
on B, so De Giorgi's local boundedness and oscillation decay apply to it. The error w is small
on small balls exactly because 2 - n/r > 0. Comparing the essential oscillation of u on
concentric balls with that of h gives
osc_{B(x₀, s)} u ≤ C₀ (s/ρ)^α₀ osc_{B(x₀, ρ)} u + C ρ^(2 - n/r) ‖f‖_{Lʳ} / λ,
and the iteration lemma TauCeti.exists_le_mul_div_rpow_mul_add_mul_rpow_of_le_mul_add turns
this into a power law with exponent min (2 - n/r) (α₀/2). The precise representative then
converts the power law into a Hölder bound, as for the homogeneous equation.
Applied on small balls, this gives continuity of weak solutions whose right-hand side is only
locally in Lʳ, tested against test functions; this is the form in which semilinear equations,
whose right-hand side depends on the solution, are met. On a general finite-dimensional inner
product space the Laplacian case follows by orthonormal coordinates.
Main declarations #
All in the namespace TauCeti.PDE.
exists_forall_energyFormH1_eq_zero_ae_abs_sub_le: on a ball, a weak solution with right-hand sidefis a weak solution of the homogeneous equation up to an error bounded byK ρ^(2 - n/r) ‖f‖_{Lʳ} / λ.exists_ae_abs_value_le_mul_add_of_memLp: local boundedness,|u| ≤ D (R^(-n/2) ‖u‖_{L²(B_R)} + R^(2 - n/r) ‖f‖_{Lʳ(B_R)} / λ)onB_{R/2}.exists_essSup_sub_essInf_le_mul_rpow_mul_add_of_memLp: one step of oscillation decay, comparing the essential oscillations ofuon two concentric balls.exists_ae_value_mem_Icc_add_mul_rpow_mul_add_of_memLp: the interior oscillation estimate.exists_holderOnWith_preciseRepresentative_of_memLp: De Giorgi's theorem with a right-hand side, the Hölder continuity of the precise representative.continuousOn_preciseRepresentative_of_forall_memLp: for a weak solution with a right-hand side inLʳon compact subsets ofΩ,r ≥ 2andr > n/2, the precise representative is continuous onΩ.exists_continuousOn_ae_eq_of_forall_integral_inner_gradient_eq: the same for Poisson's equation-Δu = fon an open subset of a finite-dimensional real inner product space.
References #
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Theorems 8.17, 8.22 and 8.24, and Lemma 8.23.
- Q. Han, F. Lin, Elliptic Partial Differential Equations, Chapter 4.
- M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Chapter III, Lemma 2.1.
Splitting off the right-hand side on a ball #
Splitting off the right-hand side on a ball. Let n ≥ 1 be the dimension and let r > 1
with r > n/2. There is K > 0, depending only on n, r and the normalization of mu, 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 solution of -∂ᵢ(aⁱʲ ∂ⱼu) = f, that is
a(u, v) = ∫_Ω f v for every v ∈ H¹₀(Ω), with f ∈ Lʳ(B(x₀, ρ)) and B(x₀, ρ) ⊆ Ω. Then on
B = B(x₀, ρ) there is a weak solution h ∈ H¹(B) of the homogeneous equation
-∂ᵢ(aⁱʲ ∂ⱼh) = 0 with
|u - h| ≤ K ρ^(2 - n/r) ‖f‖_{Lʳ(B)} / λ almost everywhere on B.
This transfers estimates for the homogeneous equation to equations with a right-hand side.
Local boundedness #
Local boundedness of weak solutions with a right-hand side. Let n ≥ 1 be the dimension
and let r > 1 with r > n/2. There is D > 0, depending only on λ, Λ, n, r and the
normalization of mu, 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 solution of
-∂ᵢ(aⁱʲ ∂ⱼu) = f with f ∈ Lʳ(B(x₀, R)). If B(x₀, R) ⊆ Ω, then almost everywhere on
B(x₀, R/2)
|u| ≤ D (R^(-n/2) ‖u‖_{L²(B(x₀, R))} + R^(2 - n/r) ‖f‖_{Lʳ(B(x₀, R))} / λ).
For f = 0 this is TauCeti.PDE.exists_ae_abs_value_le_mul_rpow_mul_sqrt_setIntegral.
Oscillation decay #
One step of oscillation decay with a right-hand side. Let n ≥ 1 be the dimension and
let r > 1 with r > n/2. There are α ∈ (0, 1] and C, K > 0, depending only on λ, Λ,
n, r and the normalization of mu, 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 solution of
-∂ᵢ(aⁱʲ ∂ⱼu) = f with f ∈ Lʳ(B(x₀, ρ)). If B(x₀, ρ) ⊆ Ω, u is essentially bounded on
B(x₀, ρ) and 0 < s ≤ ρ, then the essential oscillations osc_s and osc_ρ of u on
B(x₀, s) and B(x₀, ρ) satisfy
osc_s ≤ C (s/ρ)^α osc_ρ + K ρ^(2 - n/r) ‖f‖_{Lʳ(B(x₀, ρ))} / λ.
For f = 0 this is the oscillation decay
TauCeti.PDE.exists_ae_value_mem_Icc_add_mul_rpow_mul_sub of the homogeneous equation.
De Giorgi's interior oscillation estimate with a right-hand side. Let n ≥ 1 be the
dimension and let r > 1 with r > n/2. There are α ∈ (0, 1] and C > 0, depending only on
λ, Λ, n, r and the normalization of mu, 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
solution of -∂ᵢ(aⁱʲ ∂ⱼu) = f with f ∈ Lʳ(B(x₀, R)). If B(x₀, R) ⊆ Ω and 0 < s ≤ R/2,
then almost everywhere on B(x₀, s) the function u takes values in an interval of length
C (s / R)^α (R^(-n/2) ‖u‖_{L²(B(x₀, R))} + R^(2 - n/r) ‖f‖_{Lʳ(B(x₀, R))} / λ).
For f = 0 this is the form of
TauCeti.PDE.exists_ae_value_mem_Icc_add_mul_rpow_mul_rpow_mul_sqrt_setIntegral.
Hölder continuity #
De Giorgi's theorem with a right-hand side. Let n ≥ 1 be the dimension and let r > 1
with r > n/2. There are α ∈ (0, 1] and C > 0, depending only on λ, Λ, n, r and the
normalization of the additive Haar measure mu, such that the following holds. Let a be
measurable and uniformly elliptic on Ω with constants λ, Λ, let f ∈ Lʳ(Ω), and let
u ∈ H¹(Ω) be a weak solution of -∂ᵢ(aⁱʲ ∂ⱼu) = f, that is a(u, v) = ∫_Ω f v for every
v ∈ H¹₀(Ω). If the closed R-thickening of a set K lies in Ω, then the precise
representative of u is Hölder continuous on K with exponent α and constant
C R^(-α) (R^(-n/2) ‖u‖_{L²(Ω)} + R^(2 - n/r) ‖f‖_{Lʳ(Ω)} / λ).
The precise representative agrees with u almost everywhere on Ω
(TauCeti.W1p.ae_eq_preciseRepresentative). No regularity of the coefficients beyond
measurability is assumed.
Continuity for a locally integrable right-hand side #
Continuity of weak solutions with a locally integrable right-hand side. Let n ≥ 1 be the
dimension and let r ≥ 2 with r > n/2. Let a be measurable and uniformly elliptic on Ω, and
let u ∈ H¹(Ω) solve -∂ᵢ(aⁱʲ ∂ⱼu) = f against test functions, a(u, φ) = ∫_Ω f φ, where f
lies in Lʳ on every compact subset of Ω. Then the precise representative of u, which agrees
with u almost everywhere on Ω (TauCeti.W1p.ae_eq_preciseRepresentative), is continuous on
Ω.
No integrability of f on all of Ω is assumed. On each ball B whose closure lies in Ω, the
equation holds against all of H¹₀(B) (TauCeti.PDE.forall_energyFormH1_restrictL_eq_setIntegral,
which uses f ∈ L²(B)), and De Giorgi's theorem with a right-hand side,
TauCeti.PDE.exists_holderOnWith_preciseRepresentative_of_memLp, makes the restriction of u
Hölder continuous on the concentric ball of half the radius.
Continuity of weak solutions of Poisson's equation. Let E be a real inner product space
of finite dimension n ≥ 1 with its volume, and let r ≥ 2 with r > n/2. If u ∈ H¹(Ω) solves
-Δu = f weakly, ∫_Ω ⟪∇u, ∇φ⟫ = ∫_Ω f φ for every test function φ on Ω, where f lies in
Lʳ on every compact subset of Ω, then u agrees almost everywhere on Ω with a function that
is continuous on Ω.
This is the case a = 1 of TauCeti.PDE.continuousOn_preciseRepresentative_of_forall_memLp,
transported to E through the coordinates of an orthonormal basis.