Interior Hᵏ regularity #
Let a be a uniformly elliptic coefficient field on Ω whose entries are smooth on Ω, and let
u ∈ H¹(Ω) be a weak solution of the divergence-form equation
-∂ᵢ(aⁱʲ ∂ⱼu) = f in Ω, with f ∈ W^{k,2}(Ω),
meaning ∫_Ω ⟨∇v, a ∇u⟩ = ∫_Ω f v for every v ∈ H¹₀(Ω), with no boundary condition on u
and nothing assumed about ∂Ω. This file proves that u ∈ H^{k+2}_loc(Ω): on every open V
whose closure is compact and contained in Ω, the restriction of u is the value of an element
of W^{k+2,2}(V). Each derivative of the data buys one more derivative of the solution, so
smooth data give a solution in every H^m_loc(Ω). A constant matrix A is the special case
TauCeti.PDE.UniformlyEllipticOn.exists_value_eq_value_restrictL.
Differentiating the equation #
The induction on k rests on differentiating the equation. When A is constant, a weak
derivative of a weak solution is again a weak solution, with the derivative of the data as
right-hand side. If u solves -div(A ∇u) = f, ∂ₑu ∈ H¹(Ω) and ∂ₑf ∈ L²(Ω), then ∂ₑu
solves -div(A ∇(∂ₑu)) = ∂ₑf (TauCeti.PDE.energyFormH1_eq_setIntegral_of_hasWeakLineDerivOn).
Since weak derivatives commute (TauCeti.HasWeakLineDerivOn.comm), the derivative ∂ₑ moves
from ∂ⱼ(∂ₑu) onto the test function, so a(∂ₑu, φ) = -a(u, ∂ₑφ) = -∫ f ∂ₑφ = ∫ (∂ₑf) φ.
For a variable field the derivative also falls on a, and ∂ₑu solves
-∂ᵢ(aⁱʲ ∂ⱼ(∂ₑu)) = ∂ₑf + ∑ᵢⱼ (∂ᵢ∂ₑaⁱʲ ∂ⱼu + ∂ₑaⁱʲ ∂ᵢ∂ⱼu)
(TauCeti.PDE.energyFormH1_ofTestFunctionₗ_eq_setIntegral_of_hasWeakLineDerivOn). On a set Ω
with compact closure inside the open set where a is smooth, multiplication by the derivatives
of a preserves W^{k,2}(Ω) (TauCeti.Wkp.exists_value_ae_eq_mul_of_contDiffOn), so for
u ∈ H^{k+2}(Ω) and f ∈ W^{k+1,2}(Ω) the right-hand side lies in W^{k,2}(Ω)
(TauCeti.PDE.exists_hasWeakLineDerivOn_and_energyFormH1_eq_setIntegral_of_contDiffOn).
The induction #
The case k = 0 is interior H² regularity for locally Lipschitz coefficients
(TauCeti.PDE.UniformlyEllipticOn.exists_lowerOrder_eq_restrictL_of_locallyLipschitzOn). For the
step, fix V and an open W with closure V ⊆ W and closure W compact in Ω. By the
induction hypothesis the solution lies in H^{k+2}(W), so each ∂ᵢu lies in H¹(W) and solves
an equation on W with data in W^{k,2}(W). The induction hypothesis again puts ∂ᵢu in
W^{k+2,2}(V) for every i, and a function whose first weak derivatives lie in W^{k+2,2}(V)
lies in W^{k+3,2}(V) (TauCeti.Wkp.exists_value_eq_of_forall_hasWeakLineDerivOn).
Main declarations #
TauCeti.PDE.energyFormH1_ofTestFunctionₗ_eq_neg_of_hasWeakLineDerivOn: a weak derivative moves from the solution to the test function across the constant-coefficient energy form.TauCeti.PDE.energyFormH1_eq_setIntegral_of_hasWeakLineDerivOn: a weak derivative of a weak solution of a constant-coefficient equation solves the differentiated equation.TauCeti.PDE.UniformlyEllipticOn.exists_hasWeakLineDerivOn_and_energyFormH1_eq_neg_setIntegral: forL²data, each first derivative of a weak solution lies inH¹on compactly contained open sets and solves the equation with the distributional derivative of the data.TauCeti.PDE.exists_hasWeakLineDerivOn_and_energyFormH1_eq_setIntegral_of_contDiffOn: for smooth coefficients,u ∈ H^{k+2}and data inW^{k+1,2}, each first derivative of a weak solution solves an equation with data inW^{k,2}.TauCeti.PDE.UniformlyEllipticOn.exists_value_eq_value_restrictL_of_contDiffOn: interiorH^{k+2}regularity for smooth coefficients and data inW^{k,2}.TauCeti.PDE.UniformlyEllipticOn.exists_value_eq_value_restrictL: its constant-coefficient case.
References #
- L. C. Evans, Partial Differential Equations, §6.3.1, Theorem 2 (higher interior regularity).
- D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Theorem 8.10.
Differentiating the equation #
A derivative moves across the constant-coefficient energy form. Let A be a constant
matrix 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 a(d, φ) = -a(u, ∂ₑφ) for every test function φ on Ω.
No ellipticity is needed. This is the case ∂ₑA = 0 of
TauCeti.PDE.energyFormH1_ofTestFunctionₗ_eq_neg_sub_of_hasWeakLineDerivOn.
A weak derivative of a weak solution is a weak solution. Let A be a constant matrix and
let u ∈ H¹(Ω) satisfy a(u, v) = ∫_Ω f v for every v ∈ H¹₀(Ω), where
a(u, v) = ∫_Ω ⟨∇v, A ∇u⟩. If, in a direction e, u has a weak derivative ∂ₑu ∈ H¹(Ω) and
f has a weak derivative ∂ₑf ∈ L²(Ω), then a(∂ₑu, v) = ∫_Ω (∂ₑf) v for every
v ∈ H¹₀(Ω): the derivative ∂ₑu solves -div(A ∇(∂ₑu)) = ∂ₑf in Ω.
No ellipticity is needed, and no boundary condition is imposed on u.
The derivative of a weak solution with L² data. Let A be a constant, uniformly
elliptic matrix, f ∈ L²(Ω), and let u ∈ H¹(Ω) be a weak solution of -∂ᵢ(Aⁱʲ ∂ⱼu) = f in
Ω, in the sense that ∫_Ω ⟨∇v, A ∇u⟩ = ∫_Ω f v for every v ∈ H¹₀(Ω). Let V be open with
compact closure contained in Ω. Then in every direction e the weak derivative ∂ₑu lies in
H¹(V) and satisfies ∫_V ⟨∇φ, A ∇∂ₑu⟩ = -∫_V f ∂ₑφ for every test function φ on V: it is
a weak solution of -∂ᵢ(Aⁱʲ ∂ⱼ∂ₑu) = ∂ₑf in V, with the derivative of the data taken in the
sense of distributions.
No regularity of f beyond L² and no regularity of ∂Ω is assumed.
Differentiating the equation with smooth coefficients #
The derivative of a weak solution with smooth coefficients. Let the entries of a be
smooth on an open set U, let Ω have compact closure contained in U, let F ∈ W^{k+1,2}(Ω),
and let u ∈ H^{k+2}(Ω) be a weak solution of -∂ᵢ(aⁱʲ ∂ⱼu) = F in Ω, in the sense that
∫_Ω ⟨∇v, a ∇u⟩ = ∫_Ω F v for every v ∈ H¹₀(Ω). Then in every direction e the weak
derivative ∂ₑu lies in H¹(Ω) and is a weak solution of
-∂ᵢ(aⁱʲ ∂ⱼ(∂ₑu)) = G in Ω
for some G ∈ W^{k,2}(Ω). The differentiated equation
(TauCeti.PDE.energyFormH1_ofTestFunctionₗ_eq_setIntegral_of_hasWeakLineDerivOn) has right-hand
side ∂ₑF + ∑ᵢⱼ (∂ᵢ∂ₑaⁱʲ ∂ⱼu + ∂ₑaⁱʲ ∂ᵢ∂ⱼu), which lies in W^{k,2}(Ω) because multiplication
by the derivatives of a preserves it.
No ellipticity is needed, and no boundary condition is imposed on u.
Interior H^{k+2} regularity #
Interior H^{k+2} regularity for smooth coefficients. Let a be uniformly elliptic on
Ω, with entries smooth on Ω, let F ∈ W^{k,2}(Ω), and let u ∈ H¹(Ω) be a weak solution of
-∂ᵢ(aⁱʲ ∂ⱼu) = F in Ω,
in the sense that ∫_Ω ⟨∇v, a ∇u⟩ = ∫_Ω F v for every v ∈ H¹₀(Ω), with no boundary condition
on u. Then u ∈ H^{k+2}_loc(Ω): on every open V whose closure is compact and contained in
Ω, the restriction of u is the value of an element of W^{k+2,2}(V).
No regularity of ∂Ω is assumed, and the derivatives of the coefficients need not be bounded
on Ω.
Interior H^{k+2} regularity for a constant principal coefficient. Let A be a
constant, uniformly elliptic matrix, let F ∈ W^{k,2}(Ω), and let u ∈ H¹(Ω) be a weak solution
of
-∂ᵢ(Aⁱʲ ∂ⱼu) = F in Ω,
in the sense that ∫_Ω ⟨∇v, A ∇u⟩ = ∫_Ω F v for every v ∈ H¹₀(Ω), with no boundary condition
on u. Then u ∈ H^{k+2}_loc(Ω): on every open V whose closure is compact and contained in
Ω, the restriction of u is the value of an element of W^{k+2,2}(V). This is the constant
case of TauCeti.PDE.UniformlyEllipticOn.exists_value_eq_value_restrictL_of_contDiffOn.
No regularity of ∂Ω is assumed.