Decay estimates for elliptic equations with constant coefficients #
Let A be a constant n × n coefficient matrix whose quadratic form satisfies
λ ‖ξ‖² ≤ ξᵀ A ξ ≤ Λ ‖ξ‖² with 0 < λ, and let u be a C² solution of
∑ᵢⱼ Aᵢⱼ ∂ᵢ∂ⱼu = 0
on a ball B_R = ball x₀ R ⊆ ℝⁿ, with values in a Banach space. This file proves the two decay
estimates on concentric balls B_ρ ⊆ B_R that harmonic functions satisfy
(InnerProductSpace.HarmonicOnNhd.setIntegral_ball_norm_sq_le and
InnerProductSpace.HarmonicOnNhd.setIntegral_ball_norm_sub_setAverage_sq_le), now with constants
depending only on n and the ellipticity ratio Λ / λ:
∫_{B_ρ} ‖u‖² ≤ C (ρ / R)ⁿ ∫_{B_R} ‖u‖²,
∫_{B_ρ} ‖u - u_ρ‖² ≤ C (ρ / R)^(n + 2) ∫_{B_R} ‖u - c‖²
for every constant c, where u_ρ is the average of u over B_ρ. For a scalar solution the
gradient ∇u satisfies the same two estimates, and so does the weak gradient of a weak solution
u ∈ H¹(Ω) of -div(A ∇u) = 0 on every ball B_R ⊆ Ω.
These are the decay estimates of the frozen problem in the Campanato approach to Schauder
estimates for equations with variable coefficients: on each ball the coefficients are frozen at
the centre, and the solution is compared with a solution of the resulting constant-coefficient
equation. The Laplacian, the case A = 1, is not enough there, since the frozen coefficient
matrix is an arbitrary elliptic matrix.
The comparison itself is carried out here for a weak solution in H¹(Ω) of
-div(A ∇u) = -div G with G merely square integrable: the A-harmonic replacement w of u
on B_r has the decay above, and the energy estimate λ ‖∇(u - w)‖_{L²(B_r)} ≤ ‖G - c‖_{L²(B_r)}
measures the distance between them. For 0 < ρ ≤ r this gives
ψ(ρ) ≤ 4 D (ρ / r)^(n + 2) ψ(r) + ((4 D + 2) / λ²) ∫_{B_r} ‖G - c‖²
in dimension n, where ψ(t) = ∫_{B_t} ‖∇u - (∇u)_t‖² and D is the constant of the decay
estimate for the gradient.
The argument #
Only the symmetric part S of A enters the operator, because the second derivative of u is
symmetric. Let T = √S, the positive square root of the positive definite matrix S. By the
chain rule Δ (u ∘ T) = (∑ᵢⱼ (T Tᵀ)ᵢⱼ ∂ᵢ∂ⱼu) ∘ T
(TauCeti.laplacian_comp_toEuclideanCLE_apply), and T Tᵀ = S, so u solves the equation
exactly when u ∘ T is harmonic (TauCeti.PDE.exists_continuousLinearEquiv_laplacian_comp_eq
and TauCeti.PDE.harmonicOnNhd_comp_of_sum_smul_fderiv_fderiv_eq_zero, in
TauCeti/Analysis/PDE/Ellipticity/ConstantCoefficient.lean). Since
‖T y‖² = yᵀ A y, the map T stretches lengths by at most √Λ and its inverse by at most
1 / √λ, and the decay estimates for harmonic functions after a linear change of variables
(TauCeti.setIntegral_ball_norm_sq_le_of_harmonicOnNhd_comp) give the result with the bound
√(Λ / λ) on the condition number of T in the constant. The coordinates of ∇u ∘ T are
directional derivatives of the harmonic u ∘ T, so ∇u ∘ T is harmonic too
(TauCeti.PDE.harmonicOnNhd_gradient_comp_of_sum_smul_fderiv_fderiv_eq_zero), and the same
estimates apply to ∇u. A weak solution has a smooth representative solving the equation
classically, whose gradient is the weak gradient almost everywhere
(TauCeti.PDE.UniformlyEllipticOn.exists_contDiffOn_ae_eq_and_divMatrixGradient_eq_zero).
Main declarations #
TauCeti.PDE.setIntegral_ball_norm_sq_le_of_sum_smul_fderiv_fderiv_eq_zero: decay of theL²mass with exponentn.TauCeti.PDE.setIntegral_ball_norm_sub_setAverage_sq_le_of_sum_smul_fderiv_fderiv_eq_zero: decay of theL²mean oscillation with exponentn + 2.TauCeti.PDE.setIntegral_ball_norm_gradient_sq_le_of_sum_smul_fderiv_fderiv_eq_zero: decay of theL²mass of the gradient of a scalar classical solution, with exponentn.PDE.setIntegral_ball_norm_gradient_sub_setAverage_sq_le_of_sum_smul_fderiv_fderiv_eq_zero: decay of theL²mean oscillation of the gradient of a scalar classical solution, with exponentn + 2.TauCeti.PDE.setIntegral_ball_norm_gradient_sq_le_of_forall_energyFormH1_eq_zero: decay of theL²mass of the weak gradient of a weak solution inH¹(Ω), with exponentn.TauCeti.PDE.setIntegral_ball_norm_gradient_sub_setAverage_sq_le_of_forall_energyFormH1_eq_zero: decay of theL²mean oscillation of the weak gradient of a weak solution inH¹(Ω), with exponentn + 2.TauCeti.PDE.setIntegral_ball_norm_gradient_sub_setAverage_sq_le_add_of_forall_energyFormH1_eq: the comparison with the frozen problem for-div(A ∇u) = -div GwithG ∈ L².TauCeti.PDE.setIntegral_norm_gradient_sq_le_add_of_frozenandTauCeti.PDE.setIntegral_norm_gradient_sub_setAverage_sq_le_add_of_frozen: decay of theL²mass and of theL²mean oscillation of the weak gradient of anyu ∈ H¹(Ω), up to its energy distance from a weak solutionwof the constant-coefficient problem onB_r.
References #
- M. Giaquinta, L. Martinazzi, An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, Chapter 5 (decay estimates for constant coefficients and Schauder estimates by Campanato's method).
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Section 6.1 (reduction of a constant-coefficient operator to the Laplacian by a linear change of variables).
Decay of the L² mass for constant coefficients. Let A be a square matrix with
λ ‖ξ‖² ≤ ξᵀ A ξ ≤ Λ ‖ξ‖² and 0 < λ, and let u be a C² solution of
∑ᵢⱼ Aᵢⱼ ∂ᵢ∂ⱼu = 0 on the ball ball x₀ R ⊆ ℝⁿ, with ‖u‖² integrable there. Then for every
0 ≤ ρ ≤ R,
∫_{ball x₀ ρ} ‖u‖² ≤ 2ⁿ (Λ / λ)^(n / 2) (ρ / R)ⁿ ∫_{ball x₀ R} ‖u‖².
The constant depends only on n and the ellipticity ratio Λ / λ.
Decay of the L² mean oscillation for constant coefficients. Let A be a square matrix
with λ ‖ξ‖² ≤ ξᵀ A ξ ≤ Λ ‖ξ‖² and 0 < λ, and let u be a C² solution of
∑ᵢⱼ Aᵢⱼ ∂ᵢ∂ⱼu = 0 on the ball ball x₀ R ⊆ ℝⁿ, with ‖u - c‖² integrable there for some
constant c. Then for every 0 ≤ ρ ≤ R,
∫_{ball x₀ ρ} ‖u - u_ρ‖² ≤ C (Λ / λ)^((n + 2) / 2) (ρ / R)^(n + 2) ∫_{ball x₀ R} ‖u - c‖²
with C = 2^(3n + 10) (n + 1)²,
where u_ρ is the average of u over ball x₀ ρ. The constant depends only on n and the
ellipticity ratio Λ / λ. This is the decay estimate of the frozen problem in the Campanato
approach to Schauder estimates for equations with variable coefficients.
Decay of the gradient's L² mass for constant coefficients. Let A be a square matrix
with λ ‖ξ‖² ≤ ξᵀ A ξ ≤ Λ ‖ξ‖² and 0 < λ, and let g be a scalar C² solution of
∑ᵢⱼ Aᵢⱼ ∂ᵢ∂ⱼg = 0 on the ball ball x₀ R ⊆ ℝⁿ, with ‖∇g‖² integrable there. Then for every
0 ≤ ρ ≤ R,
∫_{ball x₀ ρ} ‖∇g‖² ≤ 2ⁿ (Λ / λ)^(n / 2) (ρ / R)ⁿ ∫_{ball x₀ R} ‖∇g‖².
Decay of the gradient's L² mean oscillation for constant coefficients. Let A be a
square matrix with λ ‖ξ‖² ≤ ξᵀ A ξ ≤ Λ ‖ξ‖² and 0 < λ, and let g be a scalar C² solution
of ∑ᵢⱼ Aᵢⱼ ∂ᵢ∂ⱼg = 0 on the ball ball x₀ R ⊆ ℝⁿ, with ‖∇g - c‖² integrable there for some
constant vector c. Then for every 0 ≤ ρ ≤ R,
∫_{ball x₀ ρ} ‖∇g - (∇g)_ρ‖² ≤ C (Λ / λ)^((n + 2) / 2) (ρ / R)^(n + 2) ∫_{ball x₀ R} ‖∇g - c‖²
with C = 2^(3n + 10) (n + 1)²,
where (∇g)_ρ is the average of ∇g over ball x₀ ρ.
Decay of the weak gradient's L² mass for constant coefficients. Let A be a square
matrix with λ ‖ξ‖² ≤ ξᵀ A ξ ≤ Λ ‖ξ‖² and 0 < λ, and let u ∈ H¹(Ω) be a weak solution of
-div(A ∇u) = 0 on an open set Ω ⊆ ℝⁿ, that is ∫_Ω ⟨A ∇u, ∇v⟩ = 0 for every v ∈ H¹₀(Ω). If
ball x₀ R ⊆ Ω, then for every 0 ≤ ρ ≤ R,
∫_{ball x₀ ρ} ‖∇u‖² ≤ 2ⁿ (Λ / λ)^(n / 2) (ρ / R)ⁿ ∫_{ball x₀ R} ‖∇u‖².
No boundary condition on u is assumed.
Decay of the weak gradient's L² mean oscillation for constant coefficients. Let A be
a square matrix with λ ‖ξ‖² ≤ ξᵀ A ξ ≤ Λ ‖ξ‖² and 0 < λ, and let u ∈ H¹(Ω) be a weak solution
of -div(A ∇u) = 0 on an open set Ω ⊆ ℝⁿ, that is ∫_Ω ⟨A ∇u, ∇v⟩ = 0 for every
v ∈ H¹₀(Ω). If ball x₀ R ⊆ Ω, then for every 0 ≤ ρ ≤ R and every constant vector c,
∫_{ball x₀ ρ} ‖∇u - (∇u)_ρ‖² ≤ C (Λ / λ)^((n + 2) / 2) (ρ / R)^(n + 2) ∫_{ball x₀ R} ‖∇u - c‖²
with C = 2^(3n + 10) (n + 1)²,
where (∇u)_ρ is the average of ∇u over ball x₀ ρ. No boundary condition on u is assumed.
This is the decay estimate of the frozen problem in Campanato's proof of the Schauder estimates
for divergence-form equations with Hölder continuous coefficients.
Comparison with the frozen problem for constant coefficients. Let A be a constant
coefficient matrix, uniformly elliptic with constants 0 < λ ≤ Λ, and let u ∈ H¹(Ω) solve
-div(A ∇u) = -div G weakly on an open set Ω ⊆ ℝ^{n+1}, that is
∫_Ω ⟨A ∇u, ∇φ⟩ = ∫_Ω ⟨G, ∇φ⟩ for every test function φ. Let ball x₀ r ⊆ Ω and let c be a
constant vector with G - c square integrable on ball x₀ r. Then for 0 < ρ ≤ r the square mean
oscillation ψ(t) = ∫_{B_t} ‖∇u - (∇u)_{B_t}‖² of the gradient satisfies
ψ(ρ) ≤ 4 D (ρ / r)^(n + 3) ψ(r) + ((4 D + 2) / λ²) ∫_{B_r} ‖G - c‖²,
with D = 2^(3n + 13) (n + 2)² (Λ / λ)^((n + 3) / 2). No regularity of G and no boundary
condition on u is assumed.
Decay of the L² mass of the gradient, by comparison. Let u ∈ H¹(Ω), let
ball x₀ r ⊆ Ω, and let w ∈ H¹(B_r) solve -∂ᵢ(Aⁱʲ ∂ⱼw) = 0 weakly on B_r for a constant
matrix A with λ ‖ξ‖² ≤ ξᵀ A ξ ≤ Λ ‖ξ‖², λ > 0. Then for 0 < ρ ≤ r
∫_{B_ρ} ‖∇u‖² ≤ 4 D (ρ / r)^{n+1} ∫_{B_r} ‖∇u‖² + (4 D + 2) ∫_{B_r} ‖∇u - ∇w‖², with
D = 2^{n+1} (Λ / λ)^{(n+1)/2}.
Decay of the mean oscillation of the gradient, by comparison. Let u ∈ H¹(Ω), let
ball x₀ r ⊆ Ω, and let w ∈ H¹(B_r) solve -∂ᵢ(Aⁱʲ ∂ⱼw) = 0 weakly on B_r for a constant
matrix A with λ ‖ξ‖² ≤ ξᵀ A ξ ≤ Λ ‖ξ‖², λ > 0. Then for 0 < ρ ≤ r the square mean
oscillation ψ(t) = ∫_{B_t} ‖∇u - (∇u)_{B_t}‖² satisfies
ψ(ρ) ≤ 4 D (ρ / r)^{n+3} ψ(r) + (4 D + 2) ∫_{B_r} ‖∇u - ∇w‖², with
D = 2^{3n+13} (n + 2)² (Λ / λ)^{(n+3)/2}.