Directional derivatives of higher-order Sobolev functions #
A function u ∈ Lᵖ(Ω) lies in W^{k+1,p}(Ω) exactly when, in every direction v, it has a weak
derivative ∂_v u lying in W^{k,p}(Ω); it suffices to check the directions of one orthonormal
basis. This is the description of the Sobolev scale by first derivatives,
W^{k+1,p}(Ω) = {u ∈ Lᵖ(Ω) : ∂ᵢu ∈ W^{k,p}(Ω) for every i}, and it is what lets a statement about
W^{k,p} be proved by induction on k one derivative at a time.
TauCeti.Wkp records the derivatives of u from the bottom up: the gradient first, and then the
weak Fréchet derivative of the previous field. The fields of ∂_v u are therefore those of u
with v substituted in the innermost slot, the one belonging to the gradient. At the first level
this substitution is A ↦ A† v, since the gradient is identified with a vector through the inner
product. Conversely, the fields of u are reassembled from those of the ∂ᵢu by inserting
⟪eᵢ, ·⟫ in the innermost slot. Neither direction uses the symmetry of weak second derivatives,
which is proved separately: since weak derivatives commute, the weak Hessian is symmetric.
No boundedness or boundary regularity of Ω is assumed.
Main declarations #
TauCeti.Wkp.exists_hasWeakLineDerivOn_value: every directional weak derivative of an element ofW^{k+1,p}(Ω)lies inW^{k,p}(Ω).TauCeti.Wkp.exists_hasWeakLineDerivOn_value_and_inner_gradient_ae_eq: the first derivatives of an element ofW^{2,p}(Ω)lie inW^{1,p}(Ω), with gradients given by the weak Hessian.TauCeti.Wkp.exists_value_eq_of_forall_hasWeakLineDerivOn: if the weak derivatives ofu ∈ Lᵖ(Ω)along an orthonormal basis lie inW^{k,p}(Ω), thenu ∈ W^{k+1,p}(Ω).TauCeti.Wkp.exists_value_eq_iff_forall_exists_hasWeakLineDerivOn: the two combined.TauCeti.Wkp.exists_value_ae_eq_of_forall_hasWeakLineDerivOn: the reverse direction for a function representative, with weak derivatives in every direction.TauCeti.W1p.exists_lowerOrder_eq_of_forall_hasWeakLineDerivOn: the second-order case in the form a difference-quotient argument supplies, withLᵖweak derivatives of the components⟪∇u, eⱼ⟫of the weak gradient.TauCeti.W1p.exists_lowerOrder_eq_of_gradient_eq_of_hasWeakLineDerivOn: the same on an open subdomainV ≤ Ω, for anyu ∈ W^{1,p}(V)whose weak gradient agrees onVwith that of a function onΩwhose gradient components are weakly differentiable.TauCeti.Wkp.inner_iteratedGradient_one_apply_ae_eq: the weak Hessian is symmetric.TauCeti.Wkp.adjoint_iteratedGradient_one_ae_eq: the weak Hessian is self-adjoint almost everywhere.
References #
- L. C. Evans, Partial Differential Equations, §5.2.2.
Substituting and inserting a direction in the innermost slot #
From W^{k+1,p} to the directional derivatives #
Directional derivatives of Sobolev functions are Sobolev. In every direction v, an
element of W^{k+1,p}(Ω) has a weak derivative which is the value of an element of
W^{k,p}(Ω).
First derivatives of a W^{2,p} function. In every direction v, an element u of
W^{2,p}(Ω) has a weak derivative ∂ᵥu which is the value of an element of W^{1,p}(Ω), whose
weak derivative in a direction w is the entry ⟪D²u w, v⟫ = ∂_w ∂ᵥu of the weak Hessian.
From the directional derivatives to W^{k+1,p} #
Sobolev functions from their directional derivatives. If, along every vector of an
orthonormal basis, u ∈ Lᵖ(Ω) has a weak derivative which is the value of an element of
W^{k,p}(Ω), then u is the value of an element of W^{k+1,p}(Ω).
The Sobolev scale by first derivatives. A function u ∈ Lᵖ(Ω) is the value of an element
of W^{k+1,p}(Ω) exactly when, in every direction, it has a weak derivative which is the value of
an element of W^{k,p}(Ω).
A function w in Lᵖ(Ω) whose weak derivative on Ω in every direction v is the value of
an element d v of W^{k,p}(Ω) agrees almost everywhere on Ω with the value of an element of
W^{k+1,p}(Ω). This is TauCeti.Wkp.exists_value_eq_iff_forall_exists_hasWeakLineDerivOn for a
function representative rather than an element of Lᵖ.
The first-order case of TauCeti.Wkp.exists_value_ae_eq_of_forall_hasWeakLineDerivOn: a
function w in Lᵖ(Ω) whose weak derivative on Ω in every direction v is a function w' v
in Lᵖ(Ω) agrees almost everywhere on Ω with the value of an element of W^{1,p}(Ω).
Second order from the components of the gradient #
Second-order weak differentiability from directional derivatives. Fix an orthonormal
basis e of E. If, for every pair of indices i, j, the scalar component ⟪∇u, eⱼ⟫ of the
weak gradient of u ∈ W^{1,p}(Ω) has a weak derivative in Lᵖ(Ω) in the direction eᵢ, then
u is the first-order part of an element of W^{2,p}(Ω).
This is how a difference-quotient argument, which produces exactly these componentwise derivatives, certifies membership in the second-order Sobolev space.
Second-order weak differentiability on a subdomain. Let w ∈ W^{1,p}(Ω) and let
V ≤ Ω be open. Suppose that the components ⟪∇w, eⱼ⟫ of the weak gradient of w along an
orthonormal basis e have weak derivatives in Lᵖ(Ω) along every eᵢ. Then every u ∈ W^{1,p}(V)
whose weak gradient agrees almost everywhere on V with that of w is the first-order part of an
element of W^{2,p}(V).
Symmetry of the weak Hessian #
The weak Hessian is symmetric. For u ∈ W^{2,p}(Ω) and directions v and w, the mixed
weak derivatives ⟪D²u v, w⟫ = ∂_v ∂_w u and ⟪D²u w, v⟫ = ∂_w ∂_v u agree almost everywhere on
Ω. Here D²u is the highest recorded weak derivative TauCeti.Wkp.iteratedGradient 1 u.
The weak Hessian is self-adjoint. For u ∈ W^{2,p}(Ω), the weak Hessian
D²u(x) = TauCeti.Wkp.iteratedGradient 1 u x is a self-adjoint endomorphism of E for almost
every x ∈ Ω.