2. Aspectos de la Astronomía relevantes en la Propuesta Didáctica 21
2.4 El brillo y color de las estrellas 25
After these introductory statements, we can now become more concrete: We will see now that the energy functional ЄΞH is elliptic on H2(Ϻ) and so its square root gives us an equivalent norm. And afterwards, we deduce corresponding results for a couple of other functionals that turn out to be continuous and coercive w.r.t. the H2(Ϻ)-norm.
4.49 Theorem For any Ϻ ∈ 𝕄𝑘bd(ℝ𝑑) of dimension 𝑘 ≤ 3 and any fixed D2 Ϻ-
unisolvent set Ξ, the functional ЄΞH(𝑓, 𝑔) is continuous and elliptic.
Proof: The first inequality ЄΞH(𝑓, 𝑔) ≤ c ∣∣𝑓∣∣H2∣∣𝑔∣∣H2 is fairly obvious, since by the
Sobolev embedding theorem stated in the appendix, we have H2(Ϻ) ↪ C(Ϻ), and so point evaluations are continuous in H2(Ϻ) and therefore bounded. So we have to work only for the other inequality and hence have to prove ellipticity of ЄΞH. This is done along the steps proposed in the proof of [105, Lem. 11.35] for the Euclidean case: By Rellich-Kondrachov’s Embedding Theorem 9.10 the embedding H2(Ϻ) ↪ H1(Ϻ) is compact in our case. Assume now that there were a sequence (𝑔𝑛)𝑛∈ℕ in H2(Ϻ) such that ∣∣𝑔𝑛∣∣
2
H2T(Ϻ)= 1 for all 𝑛 and nonetheless ∑
ξ∈Ξ
∣𝑔𝑛(ξ)∣2+ ЄH(𝑔𝑛) < 1 𝑛.
This gives in particular that ЄΞH(𝑔𝑛) approaches zero for 𝑛 →
∞
. By the compact embedding stated above, (𝑔𝑛)𝑛∈ℕ must have a convergent subsequence, say ( ̃𝑔𝑛) with limit 𝑔 ∈ H̃ 1(Ϻ). On the other hand, sinceЄH( ̃𝑔𝑛) → 0,
( ̃𝑔𝑛) is Cauchy in H2(Ϻ) for the equivalent norm ||⋅||H2 T
. Therefore it must have a limit there, and by uniqueness this limit must coincide with 𝑔, and we must havẽ ЄH( ̃𝑔, ̃𝑔) = 0. Moreover we have by definition of (𝑔𝑛)𝑛∈ℕ that 𝑔̃𝑛(ξ) → 0 for any ξ ∈ Ξ. As the Sobolev embedding theorem guarantees continuity of any function in H2(Ϻ) we get ̃𝑔𝑛(ξ) → ̃𝑔(ξ) and thus ̃𝑔(ξ) = 0. Due to our condition on Ξ we can only have 𝑔 = 0 satisfying this condition. But then we have a contradiction, as wẽ assumed ∣∣ ̃𝑔𝑛∣∣2H2
T(Ϻ)
= 1. q
4.50 Corollary For any D2Ϻ-unisolvent set Ξ and corresponding function values
YΞ, the functional ЄH(𝑓) is strictly convex, continuous and elliptic on the set
There are numerous other functionals and corresponding norms one can define. In this chapter, we will just consider a couple of further prominent and important examples, all of them either based on the Hessian or the Laplace-Beltrami operator. To analyse the latter, we need an appropriate version of the Calderon-Zygmund
inequality on compact ESMs that can be found in [57, Sect. 4] for the H2(Ϻ)-dense
subset of compactly supported smooth functions (cf. [63]) and thus generalises directly:
4.51 Proposition — Calderon-Zygmund Inequality —
Let Ϻ ∈ 𝕄𝑘cp(ℝ𝑑). Then for some c
1, c2 > 0 and any 𝑓 ∈ H 2(Ϻ) we have √ЄH(𝑓, 𝑓) ≤ c1∣∣𝑓∣∣L 2(Ϻ)+ c2 ∣∣ΔϺ𝑓∣∣L 2(Ϻ) .
With this result it is not hard to deduce the following result for compact ESMs, which allows us to consider Laplace-Beltrami instead of the Hessian:
4.52 Corollary For any ESM Ϻ ∈ 𝕄𝑘cp(ℝ𝑑) of dimension 𝑘 ≤ 3 and any nonempty
set Ξ, the Laplacian energy functional ЄΞ∆(𝑓, 𝑔) ∶= Є∆(𝑓, 𝑔) + ∑ξ∈Ξ𝑓(ξ) ⋅ 𝑔(ξ) for
Є∆(𝑓, 𝑔) ∶=
∫
Ϻ
є∆(𝑓, 𝑔) with є∆(𝑓, 𝑔) = (ΔϺ𝑓)(ΔϺ𝑔)
is continuous and elliptic.
Proof: We can bound ЄΞ∆(𝑓, 𝑔) ≤ c ЄΞH(𝑓, 𝑔) ≤ c ∣∣𝑓∣∣H2(Ϻ)∣∣𝑔∣∣H2(Ϻ), so continuity
comes directly once more, and only the other relation is of interest again. But now we have Calderon-Zygmund, and thereby we obtain
∣∣𝑓∣∣2H2 T(Ϻ) ≤ c (∣∣𝑓∣∣ 2 H1T(Ϻ)+ ∣∣ΔϺ𝑓∣∣ 2 L2(Ϻ)) =∶ ∣∣𝑓∣∣ 2 H2∆(Ϻ) by adding ∣∣𝑓∣∣2H1 T(Ϻ)
on both sides. This gives equivalence〈12〉of ||⋅||
H2(Ϻ) and the aux- illiary norm ||⋅||H2
∆(Ϻ). Consequently, we can more or less copy the proof of Theo-
rem 4.49; we just have to replace the tangential Hessian energy by the Laplacian energy if we recall that in compact ESMs a single point suffices to force a function with vanishing Laplace-Beltrami to vanish if interpolating zero. q From these functionals, one can easily deduce others and obtain continuity, strict convexity and coercivity or even ellipticity for them. In particular, we have the respective result for the functional
ЄΞ𝜂(𝑓) ∶= 𝜂 ⋅ ЄH(𝑓) + (1 − 𝜂) ⋅ ∑ ξ∈Ξ
(𝑓(ξ) − үξ)2
〈12〉In fact, one can also find in [36, Lem. 3.2] that the standard norm on H2
(Ϻ) and this auxiliary norm are equivalent, at least for the hypersurface case.
for D2Ϻ-unisolvent Ξ, 𝜂 ∈ ]0, 1[ and arbitrary (үξ)ξ∈Ξ on H2(Ϻ), provided we have dim Ϻ = 𝑘 < 4: We see continuity directly from our previous arguments and the fact that by Sobolev embedding point evaluation is continuous on H2(Ϻ) if dim Ϻ < 4; in fact, ЄΞ𝜂(𝑓) differs from ЄΞH(𝑓) only by scalar multiplication and the affine linear term
𝜂 ⋅ ∑ ξ∈Ξ
((үξ)2− 2𝑓(ξ)ү ξ) ,
which is easily seen to be continuous. Strict convexity is clear by the corresponding property of ЄΞH(𝑓), as an additional constant or linear portion will not affect this, and coercivity comes by the ellipticity of ЄΞH(𝑓).
4.53 Remark: We can actually be more general in this setting: We can replace 𝜂
by a C∞(Ϻ)-function η ∶ Ϻ → [𝑎, 𝑏] with 0 < 𝑎 < 𝑏 <
∞
and (1 − 𝜂) by pointwise weights ИΞ= {𝜂ξ}〈13〉with 0 < 𝜂ξ <
∞
for each ξ ∈ Ξ without doing any harm to the relevant properties. These choices yield different levels of detail for a smoothing or balancing process on H2(Ϻ). That process is then capable of stressing smoothness in terms of Hessian energy or approximation in a finite set of points both globally and locally, similar to the usual formulation of smoothing splines (cf. [33]). And in the course of this thesis, we will revisit this and the previous functionals to achieve indeed corresponding approximations of energy minima under point constraints and smoothing!After discussing functionals based on certain energy norms accompanied by point evaluations up to this point, we now turn to another useful functional where there are no point evaluations, but which provides nonetheless all relevant properties. This is a functional that will allow us later to solve the partial differential equation ΔϺ𝑓 − 𝜆𝑓 = 𝑔 on a compact ESM without boundary, provided this solution exists and is an element of H2(Ϻ):
4.54 Theorem The functional
Є𝜆∆(𝑓) ∶=
∫
Ϻ(ΔϺ𝑓 − 𝜆𝑓)2
is continuous and elliptic on H2(Ϻ) for any Ϻ ∈ 𝕄𝑘
cp(ℝ𝑑) and any 𝜆 > 0.
Proof: We obtain by Green’s theorem that
〈13〉The letters И,ͷ are a variant of the greek letter ”Digamma/Wau”, representing the sound ”[w]”
of standard English like in ”weight”. We use this here because apparently a sound with value ”[w]” is the first hand choice for such set of weights, and both standard latin ”W” and standard greek ”Digamma”-glyph ”Ϝ” would obviously be misleading. Additionally, И/ͷ is also the cyrillic successor of greek ”H,𝜂”, making И also the closest relative to uppercase 𝜂 from a certain point of view.
∫
Ϻ (ΔϺ𝑓 − 𝜆𝑓)2 =∫
Ϻ (ΔϺ𝑓)2− 2𝜆∫
Ϻ 𝑓ΔϺ𝑓 + 𝜆2∫
Ϻ 𝑓2 =∫
Ϻ (ΔϺ𝑓)2+ 𝜆2∫
Ϻ 𝑓2+ 2𝜆∫
Ϻ ⟨∇Ϻ𝑓, ∇Ϻ𝑓⟩which is up to weighting factors 𝜆, 𝜆2 precisely the auxiliary norm ∣∣𝑓∣∣2H2 ∆(Ϻ)
. q
As a result of this, we see that √Є𝜆∆(⋅) gives us an equivalent norm on H2(Ϻ). If we now demand that there is indeed a solution 𝑓∗ ∈ H2
(Ϻ) to the partial differential equation ΔϺ𝑓 − 𝜆𝑓 = 𝑔, then by definition this function satisfies ΔϺ𝑓∗− 𝜆𝑓∗ = 𝑔 in the H2-sense. Thus we can deduce for any 𝑓 ∈ H2 the relation
∣∣𝑓 − 𝑓∗∣∣2 H2(Ϻ)≤ c Є 𝜆 ∆(𝑓 − 𝑓∗) = c
∫
Ϻ (ΔϺ(𝑓 − 𝑓∗) − 𝜆(𝑓 − 𝑓∗))2 = c∫
Ϻ (ΔϺ𝑓 − 𝜆𝑓 − (ΔϺ𝑓∗− 𝜆𝑓∗)2= c∫
Ϻ (ΔϺ𝑓 − 𝜆𝑓 − 𝑔)2,whereby the residual of the equation is equivalent to the squared norm distance to the solution. Consequently, the solution of the equation reduces to some residual
energy minimisation.
Similarly, we can also handle the situation 𝜆 = 0, because we already know that for a given function value at a single point ξ0 (and continuous point evaluation) the energy functional Є∆(𝑓) + (𝑓(ξ0))2 is elliptic, and thus for a fixed interpolation constraint 𝑓(ξ0) = ү0 and the PDE ΔϺ𝑓 = 𝑔, the residual again turns out to be a squared norm distance to the solution 𝑓∗of the form
Є∆(𝑓 − 𝑔) + (𝑓(ξ0) − ү0)2 =
∫
Ϻ (ΔϺ𝑓 − 𝑔)2+ (𝑓(ξ0) − ү0)2 =∫
Ϻ (ΔϺ𝑓 − ΔϺ𝑓∗)2+ (𝑓(ξ 0) − 𝑓 ∗(ξ 0)) 2 ≥ c ∣∣𝑓 − 𝑓∗∣∣2 H2(Ϻ).4.55 Remark: (1) The solvability of even these simple PDEs of the form ΔϺ𝑓−𝜆𝑓 = 𝑔 with 𝜆 ≥ 0 is quite a delicate matter, particularly when regularity questions are involved — we recall in particular that we require a solution to be indeed a function from H2, while commonly the approach of choice (cf. [37, 52]) are so called weak
solutions, which can only be expected to be elements of H1. However, the general
”rule of thumb” for the regularity of a solution is that if 𝑔 ∈ Hℓ then 𝑓∗∈ Hℓ+2 (cf. [37, 52, 55]). Regarding the matter of solvability (which we shall not address any further) at least in case of the equation Δ𝑓 = 𝑔, an application of Green’s theorem shows that in our case of a compact ESM the function 𝑔 must necessarily have vanishing integral because it holds
∫
Ϻ 1 ⋅ 𝑔 =∫
Ϻ 1 ⋅ ΔϺ𝑓 = −∫
Ϻ ⟨0, ∇Ϻ𝑓⟩ = 0.(2) As usual, the presence of a boundary introduces further difficulties. In our case, it would introduce the need for boundary conditions, and we would have to include them in our functionals to maintain ellipticity in the present situation. Indeed, this is possible to some degree (cf. [55]), but unfortunately the inclusion of the bound- ary conditions into the functional is of fractional Sobolev type and therefore highly uncomfortable in practical terms.
(Ʒ) One could also look for an appropriate expression for more general intrinsic partial differential operators and equations (cf. [36, 55]), but the detailed discus- sion of these would lead too far here. So we leave the transformation of more general intrinsic PDE into our setting to the future.