Comparison with the frozen-coefficient problem #
Let u ∈ H¹(Ω) be a weak solution of the divergence-form equation
-∂ᵢ(aⁱʲ ∂ⱼu) = -div G in Ω,
tested against test functions, with a variable coefficient matrix a. On an open B ⊆ Ω, freeze
the coefficients at a constant elliptic matrix A, typically A = a(x₀) for a centre x₀, and
let w ∈ H¹(B) be the weak solution of the frozen problem -∂ᵢ(Aⁱʲ ∂ⱼw) = 0 in B with the
boundary values of u, u - w ∈ H¹₀(B). This file bounds the energy of the error u - w:
if ‖a(x) - A‖ ≤ ε on B and A has lower ellipticity constant λ, then
λ ‖∇(u - w)‖_{L²(B)} ≤ ε ‖∇u‖_{L²(B)} + ‖G - c‖_{L²(B)}
for every constant c such that G - c ∈ L²(B). For α-Hölder coefficients and data on an
open set B ⊆ B(x₀, r), freezing at x₀ makes both terms of order r^α:
λ ‖∇(u - w)‖_{L²(B)} ≤ r^α ([a]_α ‖∇u‖_{L²(B)} + [G]_α |B|^{1/2}).
This is the comparison step of Campanato's approach to the Schauder estimates for equations with
Hölder continuous coefficients. Squared, it bounds ∫_B ‖∇u - ∇w‖² by
2 λ⁻² r^{2α} ([a]_α² ∫_B ‖∇u‖² + [G]_α² |B|). Together with the decay of solutions of the frozen
problem on concentric balls
(TauCeti.PDE.setIntegral_ball_norm_sub_setAverage_sq_le_of_sum_smul_fderiv_fderiv_eq_zero), the
coefficient term plays the role of the perturbation ε in the iteration lemma
TauCeti.exists_le_mul_div_rpow_mul_add_mul_rpow_of_le_mul_add and the data term that of its
inhomogeneity. For a = 1 and A = 1 the coefficient term vanishes and the estimate is the one
behind TauCeti.PDE.exists_holderOnWith_preciseRepresentative_gradient.
The coefficient oscillation is measured in the operator norm of the attached bilinear forms
TauCeti.PDE.matrixBilinearForm, as in the Lipschitz hypothesis of whole-space H² regularity.
The argument #
Since constants have zero divergence, u solves -∂ᵢ(aⁱʲ ∂ⱼu) = -div (G - c) on B against all
of H¹₀(B) (TauCeti.PDE.forall_energyFormH1_restrictL_eq_inner). Adding -∂ᵢ((A - a)ⁱʲ ∂ⱼu)
to both sides, u solves the frozen equation
-∂ᵢ(Aⁱʲ ∂ⱼu) = -div (G - c + (A - a) ∇u) on B,
whose field is square integrable with norm at most ‖G - c‖_{L²(B)} + ε ‖∇u‖_{L²(B)}. The
energy estimate for the replacement relative to the elliptic matrix A
(TauCeti.PDE.UniformlyEllipticOn.mul_norm_gradient_sub_le) bounds λ ‖∇(u - w)‖_{L²(B)} by
the norm of that field.
Main declarations #
TauCeti.PDE.UniformlyEllipticOn.mul_norm_gradient_restrictL_sub_le: the comparison estimate for a frozen elliptic matrixAwith‖a - A‖ ≤ εonB.TauCeti.PDE.UniformlyEllipticOn.exists_sub_mem_w1p0Submodule_mul_norm_gradient_sub_le: on an open set inside a ballB(x₀, r), the replacement frozen atx₀exists and its error isO(r^α)forα-Hölder coefficients and data.TauCeti.PDE.UniformlyEllipticOn.exists_frozen_setIntegral_norm_gradient_sub_sq_le: the squared form of this estimate on a ballB(x₀, r),∫_{B_r} ‖∇u - ∇w‖² ≤ 2 (1 + ω) r^{2α} ([a]_α² ∫_{B_r} ‖∇u‖² + [G]_α² r^{n+1}) / λ², whereωis the volume of the unit ball.
References #
- M. Giaquinta, L. Martinazzi, An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, 2nd ed., Chapter 5 (Schauder estimates for equations with variable coefficients, by freezing the coefficients).
- Q. Han, F. Lin, Elliptic Partial Differential Equations, Chapter 3.
Comparison with the frozen problem. Let u ∈ H¹(Ω) solve -∂ᵢ(aⁱʲ ∂ⱼu) = -div G
weakly on Ω, tested against test functions, let B ⊆ Ω be open, and let A be a constant
matrix, uniformly elliptic with lower constant λ, with ‖a(x) - A‖ ≤ ε for x ∈ B in the
operator norm of the attached bilinear forms. If w ∈ H¹(B) has the boundary values of u,
u - w ∈ H¹₀(B), and solves the frozen problem -∂ᵢ(Aⁱʲ ∂ⱼw) = 0 weakly on B, then for every
constant c with G - c ∈ L²(B)
λ ‖∇(u - w)‖_{L²(B)} ≤ ε ‖∇u‖_{L²(B)} + ‖G - c‖_{L²(B)}.
The replacement frozen at a centre. Let u ∈ H¹(Ω) solve -∂ᵢ(aⁱʲ ∂ⱼu) = -div G weakly
on Ω ⊆ ℝ^{n+1}, tested against test functions, let B ⊆ Ω be open with B ⊆ B(x₀, r), and let
a be uniformly elliptic with constants 0 < λ ≤ Λ on a set containing x₀. Suppose that on
B(x₀, r) the coefficients are α-Hölder with constant K_a, in the operator norm of the
attached bilinear forms, and G is α-Hölder with constant K_G, where α > 0. Then there is
w ∈ H¹(B) with the boundary values of u, u - w ∈ H¹₀(B), solving the problem
-∂ᵢ(a(x₀)ⁱʲ ∂ⱼw) = 0 with coefficients frozen at x₀ weakly on B, such that
λ ‖∇(u - w)‖_{L²(B)} ≤ r^α (K_a ‖∇u‖_{L²(B)} + K_G |B|^{1/2}).
The replacement is unique, by
UniformlyEllipticOn.existsUnique_sub_mem_w1p0Submodule_forall_energyFormH1_eq_of_subset_ball.
The frozen replacement on a ball, squared. Let u ∈ H¹(Ω) solve -∂ᵢ(aⁱʲ ∂ⱼu) = -div G
weakly on Ω ⊆ ℝ^{n+1}, let ball x₀ r ⊆ Ω, let a(x₀) be uniformly elliptic with lower
constant λ, and let a and G be α-Hölder on ball x₀ r with constants K_a and K_G.
Then there is w ∈ H¹(B_r) solving the homogeneous problem -∂ᵢ(a(x₀)ⁱʲ ∂ⱼw) = 0 frozen at x₀
weakly on B_r with
∫_{B_r} ‖∇u - ∇w‖² ≤ 2 (1 + ω) r^{2α} (K_a² ∫_{B_r} ‖∇u‖² + K_G² r^{n+1}) / λ², where ω is the
volume of the unit ball. The proof takes for w the frozen replacement of
UniformlyEllipticOn.exists_sub_mem_w1p0Submodule_mul_norm_gradient_sub_le, which has the
boundary values of u; that boundary condition is not part of the conclusion.