Differentiating a divergence-form equation with smooth coefficients #
Let a be a matrix-valued coefficient field whose entries are smooth on an open Ω ⊆ ℝⁿ, and let
u ∈ H¹(Ω) be a weak solution of
-∂ᵢ(aⁱʲ ∂ⱼu) = f in Ω,
meaning ∫_Ω ⟨∇φ, a ∇u⟩ = ∫_Ω f φ for every test function φ. When the coefficients are
constant, a weak derivative ∂ₑu ∈ H¹(Ω) solves the same equation with data ∂ₑf. For variable
coefficients the derivative also falls on a, and ∂ₑu solves
-∂ᵢ(aⁱʲ ∂ⱼ(∂ₑu)) = ∂ₑf + ∂ᵢ((∂ₑaⁱʲ) ∂ⱼu) in Ω.
The commutator ∂ᵢ((∂ₑaⁱʲ) ∂ⱼu) is a distribution in general. Once u has second weak
derivatives it is the function ∑ᵢⱼ (∂ᵢ∂ₑaⁱʲ ∂ⱼu + ∂ₑaⁱʲ ∂ᵢ∂ⱼu), which lies in L²_loc(Ω) when
u ∈ H²_loc(Ω). This is the induction step of interior Hᵏ regularity for smooth coefficients:
interior H² regularity for ∂ₑu, a solution of an equation with L² data, puts u in H³,
and so on.
All three statements are identities against test functions and need neither ellipticity, nor
boundedness of the coefficients, nor a boundary condition on u. Smoothness of the coefficients
on Ω is more than the identities use (C¹, respectively C², would do), but it is what makes
the product of a coefficient with a test function a test function, so that the Leibniz rule
TauCeti.HasWeakLineDerivOn.contDiffOn_smul applies, and it is the hypothesis of the bootstrap
to C^∞.
Main declarations #
TauCeti.PDE.energyFormH1_ofTestFunctionₗ_eq_neg_sub_of_hasWeakLineDerivOn: a weak derivative∂ₑmoves from the solution onto the test function across the energy form, at the cost of the energy form of the coefficient field∂ₑa.TauCeti.PDE.energyFormH1_ofTestFunctionₗ_eq_neg_setIntegral_of_hasWeakLineDerivOn: foruwith second weak derivatives, the energy form of a smooth coefficient field against a test function is integration against the function-∂ᵢ(Bⁱʲ ∂ⱼu).TauCeti.PDE.energyFormH1_ofTestFunctionₗ_eq_setIntegral_of_hasWeakLineDerivOn: the weak derivative∂ₑuof a weak solution with second weak derivatives solves the differentiated equation with the data∂ₑf + ∑ᵢⱼ (∂ᵢ∂ₑaⁱʲ ∂ⱼu + ∂ₑaⁱʲ ∂ᵢ∂ⱼu).
References #
- L. C. Evans, Partial Differential Equations, §6.3.1, Theorem 2 (higher interior regularity), where the differentiated equation is derived in the proof.
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Theorem 8.10.
A derivative moves across a smooth-coefficient energy form. Let the entries of a be
smooth on Ω and a(u, v) = ∫_Ω ⟨∇v, a ∇u⟩. If u, d ∈ H¹(Ω) and d is the weak derivative of
u in a direction e, then for every test function φ on Ω
a(d, φ) = -a(u, ∂ₑφ) - ∫_Ω ⟨∇φ, (∂ₑa) ∇u⟩,
the last term being the energy form of the coefficient field ∂ₑa.
No ellipticity or boundedness of a is needed.
The energy form of a smooth coefficient field on an H² function. Let the entries of B
be smooth on Ω, and let u ∈ H¹(Ω) have its first weak derivatives ∂ⱼu in H¹(Ω), given as
D j ∈ H¹(Ω). Then for every test function φ on Ω
∫_Ω ⟨∇φ, B ∇u⟩ = -∫_Ω (∑ᵢⱼ (∂ᵢBⁱʲ ∂ⱼu + Bⁱʲ ∂ᵢ∂ⱼu)) φ,
where ∂ᵢ∂ⱼu = ⟪∇(D j), bᵢ⟫ and (bᵢ) is the standard basis: the energy form of B is
integration against -∂ᵢ(Bⁱʲ ∂ⱼu).
The differentiated equation for smooth coefficients. Let the entries of a be smooth on
Ω, and let u ∈ H¹(Ω) be a weak solution of -∂ᵢ(aⁱʲ ∂ⱼu) = f in Ω, in the sense that
∫_Ω ⟨∇φ, a ∇u⟩ = ∫_Ω f φ for every test function φ. Suppose that the first weak derivatives
∂ⱼu lie in H¹(Ω), given as D j, that d ∈ H¹(Ω) is the weak derivative of u in a
direction e, and that g is the weak derivative of f in the direction e. Then d solves
-∂ᵢ(aⁱʲ ∂ⱼd) = g + ∑ᵢⱼ (∂ᵢ∂ₑaⁱʲ ∂ⱼu + ∂ₑaⁱʲ ∂ᵢ∂ⱼu) in Ω
against every test function, where ∂ᵢ∂ⱼu = ⟪∇(D j), bᵢ⟫ and (bᵢ) is the standard basis.
No ellipticity or boundedness of a is needed, and no boundary condition is imposed on u.