Approximately Intrinisic Functionals

Now follow some considerations on approximately intrinsic functionals: We will apply finite dimensional function spaces defined on the ambient space in the next chapter, and we cannot really expect the resulting functions to be orthogonal ex- tensions of their restrictions in that situation; for example, this is almost hopeless for TP-splines on an ESM that is not an axis-aligned plane.

On the other hand, our functionals have corresponding formulations in terms of tangent directional Euclidean derivatives of orthogonal extensions. Our hope would be to see that if we insert just some (not necessarly orthogonal) 𝐹 with 𝑓 = Ͳ_Ϻ𝐹 in that tangent directional derivative formulation and we can make the normal derivatives approach zero, then〈14〉 we can also approximate the functional value for the function 𝑓. Or, more generally speaking, if we have a sequence of functions (𝐹_𝑛)_𝑛∈ℕ such that Ͳ_Ϻ𝐹_𝑛= 𝑓_𝑛→ 𝑓 and the normal derivatives of 𝐹_𝑛approach zero, do we also obtain that the tangent Euclidean derivative functional values for 𝐹_𝑛 approach the intrinsic functional value of 𝑓? The answer is indeed affirmative, and

〈14〉_{as was implied by Theorem 4.7 on the deviations between tangent directional and tangential}

we shall prove this now, beginning with some reconsideration of the functionals we know. Therein, we will see that they all have a common form, leading us to the concept of equivalently E_N-extrinsic functionals.

First of all, we recall that any of the specific functionals we have encountered so far has the form

Є_Ϻ(𝑓) = B0_Ϻ+ B1_Ϻ+ B2_Ϻ+ AΞ_Ϻ+ Λ_Ϻ,

where AΞ_Ϻis a possibly empty sum of (weighted) squared function values and each B𝑖_Ϻ is a possibly empty sum of (weighted) integrals taken over squared functionals of 𝑖th order tangential differentials. In particular, we have already encountered the choices B0_Ϻ(𝑓) = ∫ Ϻ 𝑓2 and B1_Ϻ(𝑓) =

∫

Ϻ ⟨∇_Ϻ𝑓, ∇_Ϻ𝑓⟩ B2_Ϻ(𝑓) =

∫

Ϻ 𝑘 ∑ 𝑖,𝑗=1 (HϺ 𝑓(τ𝑖, τ𝑗))2 or B 2 Ϻ(𝑓) =

∫

Ϻ (Δ_Ϻ𝑓)2_.

The only choice for Λ_Ϻwe encountered so far was a finite sum of function values in the functional ЄΞ_𝜂(𝑓), so that Λ_Ϻ was actually just the linear portion of AΞ_Ϻ(𝑓 − 𝑔_Y) for some suitable function 𝑔_Y.

By our theory, there is an equivalent way to express these functionals in terms of tangent directional derivatives. We suppose now that T = (τ1, ..., τ𝑘) is an or- thonormal frame of the pointwise tangent space to Ϻ that is locally smooth and C-bounded. Then we have B1_Ϻ(𝑓) = B1_T(⃡⃡⃡⃡⃡⃡𝐹) and B_Ϻ2(𝑓) = B2_T(⃡⃡⃡⃡⃡⃡𝐹) for ⃡⃡⃡⃡⃡⃡𝐹 = E_N𝑓. In our examples, we will therefore choose for the first order functional

B1_T(⃡⃡⃡⃡⃡⃡𝐹) =

∫

Ϻ 𝑑 ∑ 𝑖=1 ( 𝜕 𝜕𝑥_𝑖𝐹)⃡⃡⃡⃡⃡⃡ 2 or B1_T(⃡⃡⃡⃡⃡⃡𝐹) =

∫

Ϻ 𝑘 ∑ 𝑖=1 (D_τ𝑖𝐹)⃡⃡⃡⃡⃡⃡ 2 .

Therein, the latter choice does already present the tangential gradient version: For that choice it holds B1_T(𝐹) = B1_Ϻ(𝑓) = B1_T(⃡⃡⃡⃡⃡⃡𝐹) for any smooth 𝐹 with 𝑓 = Ͳ_Ϻ𝐹 by definition. For the second order case we will choose B2_T in correspondence to the Hessian and Laplacian energies in particular as

B2_T(⃡⃡⃡⃡⃡⃡𝐹) =

∫

Ϻ 𝑘 ∑ 𝑖,𝑗=1 (D_τ𝑖D τ𝑗𝐹)⃡⃡⃡⃡⃡⃡ 2 or B2_T(⃡⃡⃡⃡⃡⃡𝐹) =

∫

Ϻ (Δ_T𝐹)⃡⃡⃡⃡⃡⃡ 2.

As neither B0_Ϻnor AΞ_Ϻfeature any tangential derivatives, we choose just for the sake of correspondence B0_T = B0_Ϻ and AΞ_T = AΞ_Ϻ. These observations are the justification to call such functional Є_Ϻin future an equivalently E_N-extrinsic functional.

tional also for more general 𝐹 with Ͳ_Ϻ𝐹 = 𝑓; in particular, it makes sense for twice continuously differentiable functions 𝐹 ∈ C2(U_ԑ(Ϻ)). In that context, we can also consider just the corresponding tangent directional expressions B0_T, B1_T, B2_T, AΞ_T, Λ_T and obtain a functional

Є_T(𝐹) = B0_T(𝐹) + B1_T(𝐹) + B2_T(𝐹) + AΞ_T(𝐹) + Λ_T(𝐹).

The question is now what we actually loose if we take Є_Tof an arbitrary C2-function 𝐹 with Ͳ_Ϻ𝐹 = 𝑓 on the ambient space instead of the normal extension E_N𝑓. As stated before, the answer to this question can be deduced from Theorem 4.7, and we will discuss this in full detail soon, as we are going to rely heavily on this when it comes to approximation errors later.

But before we come to that, we formalise our previous observations in a couple of definitions to determine the E_N-extrinsic (energy) functionals we will actually con- sider later: Namely, we want to give a more general description of an E_N-extrinsic functional of the form Є_Ϻ(𝑓) = B_Ϻ(𝑓) + Λ_Ϻ(𝑓) + A_Ϻ(𝑓). Therein, B_Ϻ is roughly speaking the main portion of the functional, so that portion of the functional that is determined by integrals over all of Ϻ, while A_Ϻ is an auxiliary portion, like the sum over squared function values we have encountered before — but one could also think of some trace on the boundary or on a submanifold of the ESM.

To achieve this more general description, we first define a set И∞₊(Ϻ) of suitable weight functions as

И∞₊(Ϻ) ∶= ⋃ 𝑛∈ℕ

{𝑓 ∈ C∞(Ϻ, [0, 𝑛])},

so the set of all nonnegative bounded functions, and the set of all smooth bounded functions

И∞(Ϻ) ∶= ⋃ 𝑛∈ℕ

{𝑓 ∈ C∞(Ϻ, [−𝑛, 𝑛])}.

4.57 Definition We call a bilinear functional B_Ϻ on H2(Ϻ) an E_N-extrinsic total

functional if B_Ϻ(𝑓, 𝑔) = B_T(E_N𝑓, E_N𝑔) and B_T is a bilinear functional for any 𝐹, 𝐺 ∈

C2(U(Ϻ)) or 𝐹 = EN𝑓, 𝐺 = EN𝑔 with 𝑓, 𝑔 ∈ H 2

(Ϻ) that has the following form: There is some finite index set 𝐼_B such that

B_T(𝐹, 𝐺) = ∑ 𝑖∈𝐼_B

∫

Ϻ

β_𝑖(𝐹)β_𝑖(𝐺).

Each β_𝑖(𝐹) therein is of the form

where for a locally smooth and C-bounded tangent frame T = (τ1, ..., τ𝑘_{) as required} in the definition of an inverse atlas in ℿ(Ϻ) it holds for each 𝑖 ∈ 𝐼_B that

1. β0_𝑖(𝐹) = η_𝑖⋅ 𝐹 with η_𝑖∈ И∞(Ϻ), 2. β1_𝑖(𝐹) = ∑𝑘_𝑗=1η1

𝑖,𝑗⟨τ𝑗, ∇𝐹⟩ with η1𝑖,𝑗 ∈ И∞(Ϻ) for each 𝑗 = 1, ..., 𝑘, 3. β2_𝑖(𝐹) = (Λ2

𝑖(T, 𝐻𝐹)), where Λ2𝑖(T, ⋅) maps 𝑀 ∈ ℝ𝑑×𝑑 in the form Λ2 𝑖(T, 𝑀) = ∑ 1≤𝑗₁,𝑗₂≤𝑘 η2,𝑖_𝑗 1,𝑗2⋅ ((τ 𝑗₁ )t_𝑀(τ𝑗₂ )), with η2,𝑖_𝑗 1,𝑗2∈ И ∞_{(Ϻ) for any 𝑗} 1, 𝑗2 ∈ {1, ..., 𝑘}.

Directly after the following remark, we will see what the specific choices of those β_𝑖 look like in the specific energy functionals we have introduced so far.

4.58 Remark: (1) By conception, any functional of this form is in particular sym-

metric and positive semidefinite, and also continuous in H2(Ϻ).

(2) Note that the above definition depends on the locally smooth frame T. This means that in general either the transition between such frames needs to be man- aged or the ESM has to have a globally smooth T. However, for all the specific functionals we have encountered it holds everywhere on Ϻ that the term

∑ 𝑖∈𝐼_B

(β_𝑖(𝐹))2

is invariant under rotation of the frame, as it was based on the Euclidean norm of the gradient, the Frobenius norm of the Hessian or on the Laplacian, all of which are invariant under rotations. So in all those cases, we do not have to care about that issue.

4.59 Example:

1. In case of ЄΞ_H, we have that B_Tis made up of a sum of 𝑘2 purely second order functionals: For ℓ = 𝑖 + (𝑗 − 1) ⋅ 𝑘 and 𝑖, 𝑗 = 1, ..., 𝑘 each β_ℓ is of the form

β2_ℓ(𝐹) = H_𝐹(τ𝑖_{, τ}𝑗_).

2. In case of ЄΞ_𝜂, we have that B_Tcoincides with a 𝜂-multiple of that for ЄΞ_H: For ℓ = 𝑖 + (𝑗 − 1) ⋅ 𝑘 and 𝑖, 𝑗 = 1, ..., 𝑘 each β_ℓ is of the form

β2_ℓ(𝐹) = 𝜂H_𝐹(τ𝑖_{, τ}𝑗_).

3. In case of ЄΞ_∆, we have that B_Tis only made up of a single second order func- tional, so the only β2₀ takes the form

β2₀(𝐹) = 𝑘 ∑ ℓ=1

H_𝐹(τℓ_{, τ}ℓ_).

4. In case of Є𝜆_∆ on Ϻ ∈ 𝕄𝑘_cp(ℝ𝑑_{), there are two equivalent expressions: One is} to have for the only β₀(𝐹) = β2₀(𝐹) + β1₀(𝐹) + β0₀(𝐹) the choices

β2₀(𝐹) = 𝑘 ∑ ℓ=1 H_𝐹(τℓ_{, τ}ℓ_{) β}1 0(𝐹) = 0 β 0 0(𝐹) = −𝜆 ⋅ 𝐹

while the other is to have β₀ = β0₀, β_𝑘+1 = β2_𝑘+1 and for ℓ = 1, ..., 𝑘 further β_ℓ= β1_ℓ with the choices

β0₀(𝐹) = 𝜆 ⋅ 𝐹 β1_ℓ(𝐹) = √𝜆 ⋅ ⟨τℓ_{, ∇𝐹⟩ β}2 𝑘+1(𝐹) = 𝑘 ∑ ℓ=1 H_𝐹(τℓ_{, τ}ℓ_).

Now that we have formalised the main portion of the functionals we consider, we can turn to the auxilliary portion that augments the main portion.

4.60 Definition Let Ξ be a (possibly empty) finite set of points from Ϻ ∈ 𝕄𝑘_bd(ℝ𝑑) and let 𝑘 < 4 in case Ξ ≠ ⌀ to ensure continuity of point evaluations. Let {Γ_𝑖}_𝑖∈𝐼

A

be a finite (possibly empty) collection of mutually disjoint Γ_𝑖 ∈ 𝕄𝑘−1

cp (clos(Ϻ)). For weight functions η_𝑖 ∈ И∞+(Γ𝑖), 𝑖 ∈ 𝐼A, and pointwise weights 𝜂ξ ∈ ИΞ ⊆ ]0,

∞

[ we define an equivalently E_N-extrinsic augmentary functional by

A_Ϻ(𝑓, 𝑔) = ∑ ξ∈Ξ 𝜂_ξ𝑓(ξ)𝑔(ξ) + ∑ 𝑖∈𝐼_A

∫

Γ_𝑖 η_𝑖(𝑓 𝑔),

for any 𝑓, 𝑔 ∈ H2(Ϻ). We define correspondingly AT(𝐹, 𝐺) for 𝐹, 𝐺 ∈ C 2

(U(Ϻ)) or 𝐹 = ⃡⃡⃡⃡⃡⃡𝐹 = E_N𝑓, 𝐺 = ⃡⃡⃡⃡⃡⃡𝐺 = E_N𝑔 by A_T(𝐹, 𝐺) = A_Ϻ(Ͳ_Ϻ𝐹, Ͳ_Ϻ𝐺).

4.61 Remark: (1) These functionals are called ”augmentary” because they aug-

ment, so they extend, the total functionals in a certain way. Just like the latter, they are symmetric and positive semidefinite by definition. Furthermore, the trace theorem on ESMs and the definition imply that any such augmentary functional A_Ϻ(𝑓, 𝑔) is continuous on H2(Ϻ).

(2) In case that Ϻ has a nonempty boundary and Γℓ∩∂Ϻ ≠ ⌀ for some ℓ, we actually have to define with ̂Ϻ ∈ 𝕄𝑘_cp(ℝ𝑑_{) such that Ϻ ⋐ ̂Ϻ and the continuous extension} operator EϺ_Ϻ̂∶ H2(Ϻ) → H2(̂Ϻ) in the general case

∫

Γ_ℓ η_ℓ(𝑓 𝑔) ∶=

∫

Γ_ℓ η_ℓ(ͲΓ_ℓE ̂ Ϻ Ϻ𝑓)(ͲΓ_ℓE ̂ Ϻ Ϻ𝑔),

while for 𝑓, 𝑔 ∈ C(clos(Ϻ)) ∩ H2(Ϻ) we can simply presume the trace to be taken pointwise. That these different approaches yield a well-defined overall definition

can be deduced from the fact that if 𝑓1, 𝑓2coincide in H2(Ϻ) and 𝑓1 ∈ C(clos(Ϻ)), then ÊϺ_Ϻ𝑓₁ = ÊϺ_Ϻ𝑓₂ and by the trace theorem Ͳ_Γ

ℓE ̂ Ϻ Ϻ𝑓1 = ͲΓ_ℓE ̂ Ϻ Ϻ𝑓2, while on the other hand Ͳ_Γ ℓ𝑓1 is continuous.

(Ʒ) Of course, we could also include first order directional derivatives of functions on some Γ_ℓin a similar way for both tangent and Ϻ-relative (outer or inner) normal directions, but we omit these for the sake of simplicity — the necessary construc- tions are obvious. And one could also include other submanifolds of Ϻ of presum- ably higher codimension in a similar way, or relatively compact subdomains of Ϻ that are ESMs in their own right.

(4) By construction, it holds for any 𝐹, 𝐺 ∈ C2(U(Ϻ)), 𝑓 = Ͳ_Ϻ𝐹, 𝑔 = Ͳ_Ϻ𝐺 and 𝐹 = ⃡⃡⃡⃡⃡⃡𝐹 = E_N𝑓, 𝐺 = ⃡⃡⃡⃡⃡⃡𝐺 = E_N𝑔 that A_T(𝐹, 𝐺) = A_Ϻ(𝑓, 𝑔) = A_T(⃡⃡⃡⃡⃡⃡𝐹, ⃡⃡⃡⃡⃡⃡𝐺). And this relation will still hold if one of the enhancements of (Ʒ) is performed.

4.62 Example:

1. In case of ЄΞ_H, we have A_Ϻ(𝑓, 𝑔) = ∑_ξ∈Ξ𝑓(ξ)𝑔(ξ).

2. In case of ЄΞ_𝜂, we have that A_Ϻ(𝑓, 𝑔) = (1 − 𝜂) ∑_ξ∈Ξ𝑓(ξ)𝑔(ξ).

3. In case of ЄΞ_∆ on a compact Ϻ, we have again A_Ϻ(𝑓, 𝑔) = ∑_ξ∈Ξ𝑓(ξ)𝑔(ξ). 4. In case of Є𝜆∆, no AϺ is present or required. But if we would have e.g. homo-

geneous Dirichlet boundary conditions for a nonempty smooth boundary Γ of Ϻ ∈ 𝕄𝑘

sd(ℝ𝑑), then these can lead to the form A_Ϻ(𝑓, 𝑔) =

∫

Γ 𝑓𝑔.

Finally, we will formalise the linear portion of the functionals. In that, we will re- strict ourselves to linear functionals that appear as the linear portions of quadratic functionals Bℓ_Ϻ and / or Aℓ_Ϻwhen applied to 𝑓 − 𝑔_ℓ for suitable fixed 𝑔_ℓ, and speak of linear functionals subordinate to Bℓ_Ϻ and Aℓ_Ϻ.

4.63 Definition 1. Let A_Ϻ be an augmentary E_N-extrinsic functional on H2(Ϻ), based on Ξ and {Γ_𝑖}_𝑖∈𝐼

A, weights {𝜂ξ}ξ∈Ξ and weight functions {η𝑖}𝑖∈𝐼A. A linear

functional Λ_A on H2(Ϻ) is called EN-extrinsic subordinate to AϺif there are ΞΛ ⊆ Ξ and 𝐽_A ⊆ 𝐼_A such that for suitable choices of function values ү_ξ ∈ ℝ and functions 𝑔_𝑗 ∈ H32(Γ

𝑗) the functional ΛA is the linear part of ∑ ξ∈Ξ_Λ 𝜂_ξ(𝑓(ξ) − ү_ξ)2_{+ ∑} 𝑗∈𝐽_A

∫

Γ_𝑗 η_𝑗(𝑓 − 𝑔_𝑗)2_.

2. Let B_Ϻ be an E_N-extrinsic quadratic total functional on H2(Ϻ) with associated index set 𝐼_B. A linear functional Λ_B on H2(Ϻ) is called EN-extrinsic subordinate to

B_Ϻ if there is an index set 𝐽_B ⊆ 𝐼_B such that for suitable functions ⃡⃡⃡⃡⃡⃡𝐺_𝑗 = E_N𝑔_𝑗 with 𝑔_𝑗 ∈ H2(Ϻ), 𝑗 ∈ 𝐽_B, its tangent directional version Λ_B,T is the linear part of

∑ 𝑗∈𝐽_B

∫

Ϻ

β_𝑗(𝐹 − ⃡⃡⃡⃡⃡⃡𝐺_𝑗)β_𝑗(𝐹 − ⃡⃡⃡⃡⃡⃡𝐺_𝑖).

3. Finally, we say that a linear functional Λ_Ϻon H2(Ϻ) with corresponding tangent directional Λ_T is E_N-extrinsic subordinate to A_Ϻ+ B_Ϻ if it is the sum of two linear functionals Λ_Aand Λ_Bthat are either the zero functional or E_N-extrinsic subordinate to A_Ϻand B_Ϻ, respectively. This is denoted in future by Λ_Ϻ∈ 𝕃_Ϻ(A_Ϻ, B_Ϻ).

4.64 Remark: (1) Such functional is continuous on the space H2(Ϻ) by definition. (2) In particular, it is clear by this definition that there is a fixed value ԑ_ᴧ ≥ 0 such that for any 𝑓 ∈ H2(Ϻ) with ⃡⃡⃡⃡⃡⃡𝐹 = E_N𝑓 or 𝑓 = Ͳ_Ϻ𝐹, 𝐹 ∈ C2(U(Ϻ)) it holds

simultaneously that both

Є_T(⃡⃡⃡⃡⃡⃡𝐹) = B_T(⃡⃡⃡⃡⃡⃡𝐹) + Λ_T(⃡⃡⃡⃡⃡⃡𝐹) + A_T(⃡⃡⃡⃡⃡⃡𝐹) + ԑ_ᴧ ≥ 0, Є_T(𝐹) = B_T(𝐹) + Λ_Ϻ(𝐹) + A_Ϻ(𝐹) + ԑ_ᴧ≥ 0. This property is retained for any larger choice of ԑ_ᴧ.

4.65 Example:

1. In case of ЄΞ_H, we have Λ_T= 0 and therefore any ԑ_ᴧ ≥ 0 is valid. 2. In case of ЄΞ_𝜂, we have that

Λ_T(𝐹) = −2(1 − 𝜂) ∑ ξ∈Ξ

𝐹(ξ)ү_ξ,

and thus we have to require

ԑ_ᴧ ≥ (1 − 𝜂) ∑ ξ∈Ξ

(ү_ξ)2_.

3. In case of Є𝜆_∆ on a compact Ϻ, we have again Λ_T= 0 and so any ԑᴧ ≥ 0 is valid.

4.66 Definition An equivalently E_N-extrinsic Є_Ϻ = B_Ϻ+ Λ_Ϻ+ A_Ϻ with total B_Ϻ, augmentary A_Ϻ and linear Λ_Ϻ ∈ 𝕃_Ϻ(AϺ, BϺ) is called an augmented EN-extrinsic

functional and denoted by Є_Ϻ∈ 𝔼_N(Ϻ).

If Λ_Ϻ= 0, then Є_Ϻ(𝑓), Є_T(𝑓) are derived from a bilinar functional Є_Ϻ(𝑓, 𝑔), Є_T(𝑓, 𝑔) that is consequently called an augmented bilinear E_N-extrinsic functional. In that

case, we denote this fact by Є_Ϻ∈ 𝔼¤ N(Ϻ).

As stated before, the tangent directional version Є_Tof an augmented E_N-extrinsic functional Є_Ϻ= B_Ϻ+ Λ_Ϻ+ A_Ϻ∈ 𝔼_N(Ϻ) has a more general meaning for functions in C2(U(Ϻ) even if they are not normal extensions. Consequently, it is interesting

to investigate the deviation Є_T(𝐹 − ⃡⃡⃡⃡⃡⃡𝐹) for ⃡⃡⃡⃡⃡⃡𝐹 = EN(ͲϺ𝐹) and 𝐹 ∈ C 2

(U(Ϻ)), as this gives us insight into the deviation from the intrinsic functional value and will be very useful in proving upcoming results.

In doing that, we restrict ourselves first to the case Λ_Ϻ= 0 and ignore also anything in Є_Texcept for the squared purely second order terms, so the part

B2_T(𝐹) =

∫

Ϻ ∑ 𝑖∈𝐼_B

(β2_𝑖(𝐹))2.

This is possible because by construction for any 𝐹 it holds β𝑗_𝑖(𝐹 − ⃡⃡⃡⃡⃡⃡𝐹) = 0 for any 𝑖 ∈ 𝐼_B and 𝑗 = 0, 1, and by construction A_T(𝐹 − ⃡⃡⃡⃡⃡⃡𝐹) = 0. In order to simplify the notation and arguments, we suppose in the following discussion that |𝐼_B| = 1. We will see afterwards that finite summation has no relevant effect, and this restriction allows us to omit a couple of indices. So we reduce the discussion to

β2(𝐹) = ∑ 1≤𝑖,𝑗≤𝑘

η_ijH_𝐹(τ𝑖_{, τ}𝑗_).

Recalling the results of Theorem 4.7 and its proof, we can directly deduce that it holds pointwise for any 𝑥 ∈ Ϻ

∣ ∣ ∣ ∣ ∑ ij η_ij(H𝐹(𝑥)(τ𝑖, τ𝑗) − H_𝐹⃡⃡⃡⃡⃡⃡⃡(𝑥)(τ𝑖, τ𝑗)) ∣ ∣ ∣ ∣ ≤ c ∣∣∇_N𝐹(𝑥)∣∣₂.

We can thus also conclude that ∣ ∣ ∣ ∣ ∑ ij η_ij(H𝐹(𝑥)(τ𝑖, τ𝑗) − H⃡⃡⃡⃡⃡⃡⃡𝐹(𝑥)(τ 𝑖_{, τ}𝑗₎₎∣_∣ ∣ ∣ 2 ≤ c ∣∣∇_N𝐹(𝑥)∣∣2₂.

Taking integrals gives us

B2_T(𝐹 − ⃡⃡⃡⃡⃡⃡𝐹) ≤ c

∫

Ϻ

∣∣∇_N𝐹∣∣2₂.

As this works out for all summands in case |𝐼_B| > 1, we can instantly draw the following conclusion:

4.67 Conclusion For an ESM Ϻ ∈ 𝕄𝑘_bd(ℝ𝑑_{) and an augmented E}

N-extrinsic func- tional Є_Ϻ= BϺ+ AϺ∈ 𝔼¤N(Ϻ) on H

2

(Ϻ) with corresponding tangent Euclidean ex- pression Є_Tit holds for any 𝑓 ∈ C2(Ϻ), 𝐹 ∈ C2(U(Ϻ)) with 𝑓 = Ͳ_Ϻ𝐹 and ⃡⃡⃡⃡⃡⃡𝐹 = E_N𝑓 that

Є_T(𝐹 − ⃡⃡⃡⃡⃡⃡𝐹) ≤ c ∣∣∣∣∇_N𝐹∣∣₂∣∣2 L₂(Ϻ).

both B_T(𝐹), B_T(𝐺) and A_T(𝐹), A_T(𝐺) are with well-defined and finite: We can apply the usual Cauchy-Schwarz inequalities for L₂(Ϻ), for L₂(Γ_ℓ) for arbitrary ℓ or for Euclidean space (in case of Ξ ≠ ⌀) to deduce

B_T(𝐹, 𝐺) ≤ c √BT(𝐹) ⋅ √BT(𝐺), AT(𝐹, 𝐺) ≤ √AT(𝐹) ⋅ √AT(𝐺).

In particular, this allows us to deduce in the light of the previous conclusion that for any admissible choice of 𝐺 it holds

B_T(𝐹 − ⃡⃡⃡⃡⃡⃡𝐹, 𝐺) ≤ c ∣∣∣∣∇N𝐹∣∣2∣∣_L

2(Ϻ)

√B_T(𝐺), A_T(𝐹 − ⃡⃡⃡⃡⃡⃡𝐹, 𝐺) = 0. Again, we sum this up in a conclusion for future referencing:

4.68 Conclusion Let Ϻ ∈ 𝕄𝑘_bd(ℝ𝑑_{) and let 𝑓 ∈ C}2

(Ϻ). Let 𝐹 ∈ C2(U(Ϻ)) with 𝑓 = Ͳ_Ϻ𝐹 and let ⃡⃡⃡⃡⃡⃡𝐹 = E_N𝑓. Let 𝐺₁, 𝐺₂ be either in C2(U(Ϻ)) or normal extensions of functions in H2(Ϻ) for 𝑖 = 1, 2. Then it holds for any augmented E_N-extrinsic functional Є_Ϻ∈ 𝔼¤

N with ЄT= BT+ ATthat Є_T(𝐹 − ⃡⃡⃡⃡⃡⃡𝐹, 𝐺₁) ≤ c ∣∣∣∣∇_N𝐹∣∣₂∣∣

L₂(Ϻ)√BT(𝐺1).

Finally, the same arguments as applied before give us that for a linear functional Λ_B subordinate to some B_Ϻ and corresponding tangent Euclidean Λ_B,T it holds by Cauchy-Schwarz

∣Λ_B,T(𝐹 − ⃡⃡⃡⃡⃡⃡𝐹)∣ ≤ c ∣∣∣∣∇_N𝐹∣∣₂∣∣ L₂(Ϻ),

while for Λ_A this difference functional is directly clear to vanish. Thus we obtain:

4.69 Conclusion Let Ϻ ∈ 𝕄𝑘_bd(ℝ𝑑) and let BϺ+ AϺ∈ 𝔼¤N(Ϻ) with corresponding tangent Euclidean expressions B_T, A_T. Let 𝑓 ∈ C2(Ϻ), 𝐹 ∈ C2(U(Ϻ)) with 𝑓 = Ͳ_Ϻ𝐹 and ⃡⃡⃡⃡⃡⃡𝐹 = EN𝑓. Let ΛϺ∈ 𝕃Ϻ(AϺ, BϺ) be subordinate to BϺ+ AϺ. Then it holds

∣Λ_T(𝐹 − ⃡⃡⃡⃡⃡⃡𝐹)∣ ≤ c ∣∣∣∣∇_N𝐹∣∣₂∣∣ L₂(Ϻ).

Chapter 5

Ambient Functional

Approximation Methods

In this chapter we are going to revise the tangential energy functionals from an approximation theoretic point of view: Directly accessing the tangential form of

In document Ambient Approximation of Functions and Functionals on Embedded Submanifolds (Page 108-121)