H² regularity of whole-space weak solutions #
Let A be a constant, uniformly elliptic coefficient matrix and let u ∈ H¹(ℝⁿ) be a weak
solution of the divergence-form equation
-∂ⱼ(Aⁱʲ ∂ᵢu) = f on ℝⁿ, with f ∈ L²(ℝⁿ),
meaning ∫ ⟨A ∇u, ∇v⟩ = ∫ f v for every v ∈ H¹(ℝⁿ). This file proves that u then has
second-order weak derivatives in L²(ℝⁿ), that is u ∈ H²(ℝⁿ), with the estimate
‖∂_w ∂_y u‖_{L²} ≤ ‖y‖ ‖w‖ ‖f‖_{L²} / λ
in every pair of directions, where λ is the ellipticity constant. No smoothness of f and no
regularity of u beyond H¹ is assumed: constancy of the coefficient matrix, together with
ellipticity, upgrades one weak derivative to two.
The difference-quotient method #
The proof is the classical difference-quotient argument. For a direction w and a step t, the
difference quotient Dᵗ u = t⁻¹ (u(· + t w) - u) again lies in H¹(ℝⁿ), and because A is
constant the energy form is anti-adjoint for it,
a(Dᵗ u, v) = -a(u, D⁻ᵗ v)
(TauCeti.PDE.energyFormH1_differenceQuotient_eq_neg), which is the integrated form of the
discrete integration-by-parts identity ∫ (Dᵗ g) h = -∫ g (D⁻ᵗ h). Testing the equation against
Dᵗ u itself and using ellipticity on the left and the difference-quotient bound
‖D⁻ᵗ g‖_{L²} ≤ ‖w‖ ‖∇g‖_{L²} on the right gives
λ ‖∇Dᵗ u‖²_{L²} ≤ a(Dᵗ u, Dᵗ u) = -∫ f · D⁻ᵗ(Dᵗ u) ≤ ‖f‖_{L²} ‖w‖ ‖∇Dᵗ u‖_{L²},
so ‖Dᵗ ∇u‖_{L²} ≤ ‖w‖ ‖f‖_{L²} / λ uniformly in t
(TauCeti.PDE.UniformlyEllipticOn.norm_gradient_differenceQuotient_le). The
difference-quotient criterion
TauCeti.exists_norm_le_hasWeakLineDerivOn_of_frequently_eLpNorm_inv_mul_sub_le converts that
uniform bound into a weak derivative of ∇u in L², and assembling the directions of an
orthonormal basis produces a weak Fréchet derivative of ∇u, that is, the Hessian.
Working on the whole space is what keeps the argument free of cut-offs: no boundary regularity
is involved, H¹₀(ℝⁿ) = H¹(ℝⁿ) (TauCeti.w1p0Submodule_top_eq_top), so the solution concept
TauCeti.PDE.IsWeakSolutionDirichlet imposes no boundary condition here, and every difference
quotient is a legitimate test function. For a constant principal coefficient and bounded
measurable lower-order coefficients, interior H² regularity on a general domain follows by
absorbing the lower-order terms into the forcing and localizing with a cutoff
(TauCeti.PDE.UniformlyEllipticOn.exists_lowerOrder_eq_restrictL); allowing variable Lipschitz
coefficients needs the difference-quotient estimate itself to be localized.
Main declarations #
TauCeti.PDE.energyFormH1_translate: a translation moves from one argument of the constant-coefficient energy form to the other.TauCeti.PDE.energyFormH1_differenceQuotient_eq_neg: the discrete integration-by-parts identity for the constant-coefficient energy form.TauCeti.PDE.UniformlyEllipticOn.norm_gradient_differenceQuotient_le: the uniform bound on the difference quotients of the gradient of a weak solution.TauCeti.PDE.UniformlyEllipticOn.exists_norm_le_hasWeakLineDerivOn_gradient: the second-order weak directional derivatives of a weak solution, with theH²estimate.TauCeti.PDE.UniformlyEllipticOn.exists_lowerOrder_eq: a weak solution lies inH²(ℝⁿ).
References #
- L. C. Evans, Partial Differential Equations, §6.3.1, Theorem 1 (interior
H²regularity). - D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, §8.3 and Lemma 7.23.
Translation moves across the constant-coefficient energy form. For opposite vectors
h and k, translating the first argument by h is the same as translating the second by
k. This is the integrated form of the substitution x ↦ x - h, and it is where constancy of
the coefficient matrix is used.
Discrete integration by parts for the constant-coefficient energy form. Testing the
difference quotient of u against v is the same, up to sign, as testing u against the
difference quotient of v with the opposite step:
a(Dᵗ u, v) = -a(u, D⁻ᵗ v).
This is the integrated form of ∫ (Dᵗ g) h = -∫ g (D⁻ᵗ h), and it is what lets a
difference-quotient argument move the extra derivative onto the test function.
The difference quotients of the gradient of a weak solution are uniformly bounded. For a
constant, uniformly elliptic A and a weak solution u ∈ H¹(ℝⁿ) of -∂ⱼ(Aⁱʲ ∂ᵢu) = f,
‖∇Dᵗ u‖_{L²} ≤ ‖w‖ ‖f‖_{L²} / λ
for every direction w and every step t, the bound being independent of t. Since
∇Dᵗ u = Dᵗ ∇u, this is the uniform difference-quotient bound on the gradient that the
difference-quotient criterion turns into a second weak derivative.
On the whole space H¹₀(ℝⁿ) = H¹(ℝⁿ), so the hypothesis imposes no boundary condition; it is
exactly the weak equation tested against every H¹(ℝⁿ) function.
The second-order weak directional derivatives of a whole-space weak solution. For a
constant, uniformly elliptic A and a weak solution u ∈ H¹(ℝⁿ) of -∂ⱼ(Aⁱʲ ∂ᵢu) = f, the
derivative ∂_y u = ⟪∇u, y⟫ is again weakly differentiable in every direction w, with
‖∂_w ∂_y u‖_{L²} ≤ ‖y‖ ‖w‖ ‖f‖_{L²} / λ.
This is the H² estimate in quantitative, direction-by-direction form; λ is the ellipticity
constant and no other feature of A enters the bound.
A whole-space weak solution lies in H²(ℝⁿ). For a constant, uniformly elliptic A, a
weak solution u ∈ H¹(ℝⁿ) of -∂ⱼ(Aⁱʲ ∂ᵢu) = f with f ∈ L²(ℝⁿ) is the first-order part of an
element of W^{2,2}(ℝⁿ): its weak gradient is again weakly differentiable, with L² derivative.
The quantitative form of the statement, with the ellipticity constant made explicit, is
TauCeti.PDE.UniformlyEllipticOn.exists_norm_le_hasWeakLineDerivOn_gradient.