I've got a question concerning the proof of Lemma 4.2.3(c). It appears that you use the causality of Fν, but I don't really see why you use it. My argument would look as follows: The set of simple functions with compact support(!) is dense in L2,ν(R;H0)∩L2,η(R;H0), i.e., for all f∈L2,ν(R;H0)∩L2,η(R;H0) there exists a sequence (fn) of simple functions with compact support such that fn→f both in L2,ν(R;H0) and in L2,η(R;H0). This is what you show in the first 5 lines of the proof of Lemma 4.2.3(c). But from this the assertion is clear since Fν and Fη are continuous, and they coincide (by definition) on the simple functions with compact support.
Am I mistaken?
Best wishes, Hendrik
Sascha Trostorff, 2019/11/20 23:15
Dear Hendrik,
yes, you are right. Thanks for pointing out.
Best regards
Sascha
Fabian Gabel, 2019/11/17 22:45
Dear ISEM-team,
in regard to Theorem 4.2.1(d) I am tempted to say that the solution operator that maps f∈L2,ν to the solution vν,f is causal. But we only defined 'causal' for Lipschitz continuous mappings. Is the above solution operator causal? Do we have continuous dependence of the solution on the data f at all? (sorry for the naive questions)
I furthermore have a little remark on the proof of Theorem 4.3.3:
p.46, l.-7: I think it should read “by continuity of f, u0 and u.
thanks for your question. Yes, the solution operator in Theorem 4.2.1 (d) is Lipschitz-continuous, but we forgot to work that out. The argumentaion is quite simple. Choosing ν>L and uν,vν to solutions for the source terms f,g, we obtain
‖uν−vν‖=‖∂−1t,ν(Fν(uν)−Fν(vν))‖+‖∂−1t,ν(f−g)‖≤Lν‖uν−vν‖+1ν‖f−g‖
and thus,
‖uν−vν‖≤(1−L/ν)−11ν‖f−g‖.
Also thanks for spotting the missing “continuity” on p. 46.
Best regards
Sascha
Fabian Gabel, 2019/11/15 15:44
Dear ISEM-team,
working through the lecture notes, I gathered the following remarks
Remark on the Proof of Theorem 4.1.1
It may be a matter of taste but reading the proof the first few times I got confused by the way the letter ψ was used:
p.39, l.3: scalar functions in C∞c(R),
p.39, l.9: H-valued functions in dom(∂t,ν).
Questions and Remarks on the Proof of Theorem 4.2.4
p.44, l.3: “Furthermore, v is continuous …. As such, we obtain”. As you write it, “as such” suggests, that the equality in l.5 follows from the continuity of v, but I think that it is a consequence of the continuity of f, F and P: If I'm not mistaken, we approximate v∈dom(∂t,ν) by vn∈S(R;Kn) which leads to
v=∂−1t,νFν(v)=IνFν(v)=limn→∞IνF(vn)=limn→∞∫⋅−∞1[0,δ′)(τ)f(τ,P(vn(τ)))dτ=…
Does this make sense?
p.44, l.-14: “is the unique solution point of the equation of”. Do we want to use Theorem 4.2.1(b) here? Then, I would suggest to write: “is the unique fixed point of the equation w=…”
Best wishes,
Fabian
Marcus Waurick, 2019/11/15 16:02
Dear Fabian,
thank you very mich for your comments.
On page 39, you are right, we should be more consistent.
Concerning the phrase `as such', I don't think we are in the position to argue with a law professor.
The continuity of v is a consequence of the Sobolev embedding theorem: v∈dom(∂t,ν). The continuous differentiability of v then follows from the continuity of the integrand on the right-hand side f(⋅,Pv(⋅)).
Concerning your remark on p44, l -14: This is a misprint. We meant unique solution. The problem with your phrase is that an equation does not really have a fixed point.
Hope this helps (and I also hope I have addressed all your question correctly),
cheers,
MM
Fabian Gabel, 2019/11/15 20:20
Thank you Marcus for your reply and the clarifications.
As for my calculations to justify the first equality on p.44, l.5: Did these calculations make sense or were they unnecessary? We only defined F on S(R,Kn), therefore I tried to carry out an approximation which relies on the continuity of f, F and P.
In my opinion, the text suggests that the series of equations is a mere consequence of the continuity of v. As you pointed out, this is not the case.
Best wishes,
Fabian
Sascha Trostorff, 2019/11/16 14:07
Dear Fabian,
no, the computation has to be done, but as it is quite standard, we did not write it down. However, by the approximation the formula just holds almost everywhere, and then continuity gives the equality for each t. I hope this answers your question.
Best regards
Sascha
Fabian Gabel, 2019/11/17 11:06
Yes, this answers my question. Thank you very much.
Best wishes,
Fabian
Sebastian Bechtel, 2019/11/14 16:45
Dear all,
here are some more comments from Darmstadt that are not already posted at an existing suitable position in this forum ;)
But first of all, let me say in general that we had sometimes the feeling that arguments lack help in the relevant details whereas obvious calculations are carried out with more heart. You'll probably see what we mean by this in some of the following points.
Proof of Prop. 4.1.1, second last displayed formula: In the middle term the classical derivative appears. After the displayed formula you claim that you can extend this identity to dom(∂t,ν) on using that you showed it on a core. This extension works for the left-hand and right-hand sides of the identity, which is also the relevant part, however not for the whole line as claimed.
Corollary 4.1.3: The subscript H at the norm is missing.
Definition of uniformly Lipschitz continuous: First of all, we didn't like the formulation “F considered in …”. You mean that we should consider the mapping as a relation, correct? Albeit you have introduced relations in general in Lecture two, this point of view seems not so helpful here since for example you have not introduced closability in this case. So we recommend just to consider F as a mapping between subsets of metric spaces and do the Lipschitz extension in that context.
Moreover, we wondered why the definition is done in such a seemingly complicated why. By definition of Bochner measurability, the definition turns out to work quite well but on the first sight it seems unnatural. Wouldn't it be a better definition to just say that a uniformly Lipschitz continuous function is a family of Lipschitz mappings Fν:L2,ν→L2ν for ν≥μ that are consistent and satisfy supν≥μ‖Fν‖Lip? Then, starting from simple functions could be a Lemma on how to construct uniformly Lipschitz continuous functions.
Proof of Theorem 4.2.1 first part: In the third line of the proof you say that the contraction property follows from Lemma 3.2.1. This is only half the truth. First, the relevant norm bound for ∂−1t,ν is calculated in the definition after that Lemma. Second, this is the point in the proof where the assumption ν>L is used, so you should point that out.
It was not clear to us what line 6 should tell us and the formulation “can be proved in Exercise” is also a bit strange.
In the second last line you use the term “strict contraction”, it seems like there is no difference to the term “contraction” used before?
Remark 4.2.2 We where confused by the change of rolls between ν and η in the two cases. Maybe its better understandable if their ordering is fixed before entering the cases?
Lemma 4.2.3:
1) In assertion (b) you have the additional constraint that ν>0. This comes from the fact that only then ∂−1t,ν is causal. In the proof you only write that this follows directly from (a)…
3) Above the first displayed formula you have these arguments in which spaces 1(−∞,a]⋅x is and in which spaces fn−gn converges to f−g. First, you should indicate the arguments for the convergence of fn−gn (they have different roles, which results in a function with support near +∞, which decides with embedding from Remark 4.2.2 to use). While doing this, it turns out that the constraint η≥0 is not needed for this argument, in contrast to what is indicated in the text. Then you tell that with the correct combined constraints for η the following calculation follows, but it is not indicated where the previously shown assertions are used. This happens at the first limit, which is not at all explicitly explained. The arguments to justify this limit are straightforward, but you should at least tell the reader to use Hölder's inequality with the previous assertions to point into the right direction to do the standard arguments on its own.
4) In the last estimate of the first displayed formula you use the (to us) long forgotten assumption (:P) that f and g coincide on (−∞,a] together with the embeddings from the Remark. A hint that this assumption is used here would be helpful.
5) After the first displayed formula you let η→∞ which amounts to show that the integral with which one has started vanishes However, this doesn't show that Fν(f)=Fν(g). We suggest to do the calculation with 1A for all measurable subsets A⊆(−∞,a] instead of only with the maximal indicator function to get the claim.
Proof of Theorem 4.2.4: In the last displayed formula on p. 43, the δ should be δ′.
Lemma 4.3.1: In addition to what Jürgen has already mentioned: We suggest to distinguish between the equivalence part and the “Moreover”-part. In the first we suggest to introduce the parameter μ∈R as Jürgen said, but one should also quantify ν in the “Moreover”-part and the quantification should be over R, though in the proof in the last line the supremum should be taken over ν≥μ.
Best, Sebastian
Sascha Trostorff, 2019/11/14 20:43
Dear Sebastian,
thanks a lot for your remarks and suggestions. Let me comment on them:
Proof of Prop. 4.1.1, second last displayed formula: In the middle term the classical derivative appears. After the displayed formula you claim that you can extend this identity to dom(∂t,ν) on using that you showed it on a core. This extension works for the left-hand and right-hand sides of the identity, which is also the relevant part, however not for the whole line as claimed.
Of course you are right and just the term on the left- and on the right-hand side can be extended to dom(∂t,ν).
Definition of uniformly Lipschitz continuous: First of all, we didn't like the formulation “F considered in …”. You mean that we should consider the mapping as a relation, correct? Albeit you have introduced relations in general in Lecture two, this point of view seems not so helpful here since for example you have not introduced closability in this case. So we recommend just to consider F as a mapping between subsets of metric spaces and do the Lipschitz extension in that context.
I think this is a matter of taste. In general, mappings (linear or non-linear) are nothing but relations, so I do not see why our formulation is worse than yours. And yes, we did not consider closabilty in this case but we thought that taking closures of Lipschitz-continuous functions is quite standard (although, it is maybe not standard to note that the continuation is actually nothing but taking the closure).
Moreover, we wondered why the definition is done in such a seemingly complicated why. By definition of Bochner measurability, the definition turns out to work quite well but on the first sight it seems unnatural. Wouldn't it be a better definition to just say that a uniformly Lipschitz continuous function is a family of Lipschitz mappings Fν:L2,ν→L2ν for ν≥μ that are consistent and satisfy supν≥μ∥Fν∥Lip? Then, starting from simple functions could be a Lemma on how to construct uniformly Lipschitz continuous functions.
Again I would say that this is a matter of taste. However, let me emphasise that it is important to define “consistent” in your definition properly. So, for instance, it is not enough to say the the functions Fν coincide on a subset D dense in each L2,ν, since then causality would not follow.
It was not clear to us what line 6 should tell us and the formulation “can be proved in Exercise” is also a bit strange.
Well, okay. Maybe one should write that differentiability follows by Exercise 4.1 and the fact that Fν(uν) is continuous.
In the second last line you use the term “strict contraction”, it seems like there is no difference to the term “contraction” used before?#
Of course not. That was just sloppy writing. Thanks for pointing out.
Remark 4.2.2 We where confused by the change of rolls between ν and η in the two cases. Maybe its better understandable if their ordering is fixed before entering the cases?
Yes, thank you!
Lemma 4.2.3:
1) That is one way to see it. The other way would be to say that ∂−1t,νF is only a uniformly Lipschitz-continuous mapping, if ν>0.
2) Yes, this should be H0, thank you.
3) I agree, that the arguments are maybe too short. We apologise for this. And thanks for pointing out, that η≥0 is not needed for the convergence of fn−gn.
4) You are right. In the final notes we will follow the proof suggested by Hendrik, which is more structured.
5) Correct, thanks!
Proof of Theorem 4.2.1 second part: First: To go from the first to the second line in the first displayed formula you use again that ∂−1t,ν is causal (cf. 4.2.3 (b)).
Proof of Theorem 4.2.4: In the last displayed formula on p. 43, the δ should be δ′.
Thanks!
Lemma 4.3.1: In addition to what Jürgen has already mentioned: We suggest to distinguish between the equivalence part and the “Moreover”-part. In the first we suggest to introduce the parameter μ∈R as Jürgen said, but one should also quantify ν in the “Moreover”-part and the quantification should be over R, though in the proof in the last line the supremum should be taken over ν≥μ
You are right, we will rewrite it.
Best regards
Sascha
Sebastian Bechtel, 2019/11/21 16:31
Dear Sascha,
concerning consistency: My definition is what you check in Theorem 5.3.5 in the next lecture.
Best, Sebastian
Sascha Trostorff, 2019/11/21 18:08
Dear Sebastian,
you are right, however there might still be examples, where you find a nice expression for F on simple functions and not on the L2 spaces (like for the Bochner integral or for the Fourier transformation), where it seems to be more natural to define the function on that set and consider the extensions of this mapping. Anyway, it does not make any difference in the arguments need to be applied.
Best regards
Sascha
Sebastian Bechtel, 2019/11/21 18:11
Dear Sascha,
I agree that there are cases in which the simple functions are a better starting point, my ´question´ was more or less the suggestion to start with a concise definition and put this special construction in a lemma then directly making it the definition, but that´s a matter of taste and as I said, in your usage the definition doesn´t seem so unnatural afterwards.
Best, Sebastian
Sascha Trostorff, 2019/11/13 14:02
Dear all,
there is a mistake in Exercise 4.7 (b). One also has to require that for each a∈R and f,g∈D=dom(F) with f=g on (−∞,a] it follows that F(f)=F(g) on (−∞,a] (so, F is causal on D).
Best regards
Sascha
Nathanael Skrepek, 2019/11/13 12:12
Dear virtual lectures,
just before Lemma 4.3.4 you define
u(.):R∋t↦(R≤0∋θ↦u(t+θ)∈H),
which can be can be really confusing. Especially because you probably wanted it to be the other way round. I think it should be a family in θ∈R≤0 of functions in t∈R, but as it is written I read it as a family in t of functions in θ.
I would recommand to avoid this completely, by defining
uθ:{R→H,t↦u(t+θ),
for θ∈R≤0.
Jürgen Voigt, 2019/11/09 00:08
Some minor comments
Dear virtual lecturers,
Proposition 4.1.1: The final statement is (minimally) confusing, because g doesn't appear in (i). The remedy would be to start out with f,g∈L2,ν, and (i) should be f∈dom(∂t,ν), ∂t,νf=g.
Theorem 4.1.2: `representer' should be `representative'. And I do not at all understand the last line of the proof.
Line below Corollary 4.1.3: Postpone a proof would mean to do it later (I think).
Theorem 4.1.2(a): I simply do not know what `continuously differentiable at' a point means. As you do not prove this part, it does not become clear from the proof.
Lemma 4.2.3(c), misprint: The `μ' in the last space should be `η'.
Lemma 4.3.1: `uniformly Lipschitz continuous' is defined in connection with a μ∈R; see the definition on p.40. But here no μ is indicated.
4.5 Comments, line 1: you probably mean ODE (or not?).
And a comment on style: `obtain that' is verboten.
Best wishes, Jürgen
Jürgen Voigt, 2019/11/09 15:09
Dear virtual lecturers,
in the proof of Proposition 4.1.1 you quote Prop. 3.2.4 and Cor. 3.2.6, and in both of them one has the hypothesis ν≠0, whereas Prop. 4.1.1 is stated for all ν∈R. Okay, the last statement in Cor. 3.2.7 just says that Cor. 3.2.6 also holds for ν=0. Probably one should mention somewhere that the definition of ∂t,0 implies that Prop. 3.2.4 is also true for ν=0.
Best wishes, Jürgen
Marcus Waurick, 2019/11/11 08:44
Dear Jürgen,
your are completely right! Thanks!
Best regards,
moppi
Sebastian Bechtel, 2019/11/14 15:33, 2019/11/14 15:34
A further comment on the references: I think you mean DH (tensorizations of C∞c(R) with H) instead of C∞c(R;H). The latter (which is the version in the notes) would require to reference Prop. 3.2.3 instead as far as I see.
Best, Sebastian
Marcus Waurick, 2019/11/11 08:41
Dear Jürgen,
thank you for your remarks!
Your comment on Prop 4.1.1 is very helpful. Concerning Theorem 4.1.2; the word representer does exist and has also been used in this context. It means the same as representative, anyway. I'll ask the native speakers in Glasgow, whether either of these words is fine, though. I will do the same for `postponed'.
The statement in Theorem 4.1.2(a) should rather read `continuously differentiable in a neighbourhood of a'.
Your are right concerning Lemma 4.3.1; we have not been precise there. One can choose any μ.
We actually mean PDE. I have not seen many ODE approaches using Lp-spaces. We want to stress that the developed ODE theory has similarities to PDE theory, where you generalise differentiability and/or solution concepts. A posteriori one tries to find out, whether one can show that the PDE is satisfied point wise or whether the derivative operators are actually classical derivatives. For instance in the case of Poisson's equation on a bounded domain. This may be solved by an application of Riesz-representation theorem in L2 and afterwards one shows given a suitably regular right-hand side, whether the solution is smoother so that the applied Laplace may even be applied point wise to the solution.
Also thank you for the style comment.
Best regards,
moppi
Michael Doherty, 2019/11/12 14:51
Dear Jürgen,
I have some comments and questions on some of the stylistic points you raised on the lecture.
“Theorem 4.1.2: `representer' should be `representative'. And I do not at all understand the last line of the proof.”
I had to do a double take on this; I had originally thought that the two were synonymous with one another, it turns out that they are not. It's funny that they pertain to tantamount the same entity, but representer (or representor) only applies to a living person in a particular capacity. I need to remember this point, thanks!
“Line below Corollary 4.1.3: Postpone a proof would mean to do it later (I think).”
Is your point that despite the virtual lecturers writing that the proof is 'postponed' till later, they don't, by the close of the lecture, provide a proof in writing themselves? I think that the usage of 'postpone' still works in this context because, as we have seen with the comments, suggestions and exercise attempts through the previous sections, the ISem is a kind of “living” lecture series, where the insights, feedback and exercise attempts from all participants go on to form the material of the series.
“And a comment on style: `obtain that' is “And a comment on style: `obtain that' is verboten.”verboten.”
Can you clarify as to why this turn of phrase is forbidden?
Cheers,
Michael
Jürgen Voigt, 2019/11/12 19:44
Dear Michael,
I understand that I'm on dangerous ground here, not being a native speaker in contrast to you.
Concerning `representer', I was under the impression that the word doesn't even exist, and my suspicion was confirmed by its non-existence at LEO. Meanwhile I found that it does exist, but anyway, I had never seen it in the present context.
Concerning `postpone', I still would expect that something that is postponed will be treated in the text by the authors, but OK, it may be disputable. But maybe I can ask you whether the formulation `The proof is delegated to Exercise 4.2' would be acceptable for expressing that it will be posed as an exercise.
Concerning `obtain that', let me say that in mathematical German one can formulate “Von Theorem .. erhalten wir, dass v(0)=0 ist.” Translating word by word one obtains `By Theorem .. we obtain that v(0)=0.', and I have been told that `obtain' can only be used in connection with nouns. One can obtain things or results, for instance. Correct?
Best wishes, Jürgen
Michael Doherty, 2019/11/13 15:45
Dear Jürgen,
On the representer point, you were actually correct that its usage here was not right - so you actually taught me something here! (It's funny to note that both representer and representative effectively relate to the same thing; it's just that the former applies only to persons!).
Sure, I think that within the “living” or “evolving” text perspective its usage holds up. I think that the issue with “delegate”, much like with “representer” before, is that they only apply to persons. For instance, one talks about delegating a task to another person.
Cheers again for pointing these things out - they can only help to better one's style in presenting mathematics.
Best wishes,
Michael
Hendrik Vogt, 2019/11/13 22:53
Dear Michael,
and isn't it funny who asked that question on stackexchange?
Best wishes, Hendrik
Michael Doherty, 2019/11/14 11:11
Dear Hendrik,
Funny indeed!
Cheers,
Michael
Jürgen Voigt, 2019/11/13 14:38
Dear Michael,
incidentally, I also found a dangling participle on p. 42, line -4.
Best wishes, Jürgen
Michael Doherty, 2019/11/14 11:23
Hi Jürgen,
Do you mean the sentence: “Hence, again using Lemma 4.2.3, it follows that”? If so, well spotted!
Cheers,
Michael
Sebastian Bechtel, 2019/11/14 15:47
Dear Jürgen,
“And I do not at all understand the last line of the proof.”
I agree that this is hardly understandable from the written text. The given calculation shows the norm estimate without using anything else and for the continuity one has to redo the calculation using f(t)−f(tn), which amounts to restrict the range of integration in the calculation. Clearly, the shown calculation then gives a dominating function (1(−∞,t+ε)‖g‖) and we can use the dominated convergence theorem to conclude continuity.
Best, Sebastian
Sascha Trostorff, 2019/11/14 16:18
Dear Sebastian, dear Jürgen,
the reference to dominated convergence is made to see that t↦∫t−∞g(s)ds is continuous for g∈L2,ν. However, I do not see why you have to redo the calculation as Sebastian indicated.
Best regards
Sascha
Jürgen Voigt, 2019/11/14 17:04
Dear Sascha,
OK; now I understand that you use the estimate to see that g is integrable over intervals (−∞,a] for all a∈R, and then you also use the definition of f. (This is not reflected by the sequence of statements, in particular, because the estimate starts with the estimate for ‖f(t)‖). And why do you need dominated convergence for the desired norm estimate? (The given estimate is quite sufficient for the assertion.)
Best wishes, Jürgen
Sascha Trostorff, 2019/11/14 19:13
Dear Jürgen,
of course you don’t need dominated convergence for the estimate but just for the continuity. The formulation in the text is misleading, sorry for that.
Best regards
Sascha
Sebastian Bechtel, 2019/11/14 22:22
Dear Sascha,
you are right, one doesn't have to redo the whole calculation. But at least the first step. Maybe it would be better to first show the integrability of ‖g‖ over intervals with the respective estimate and then use this to conclude the estimate on the one hand and secoundly perform the continuity argument using dominated convergence and the previously established dominating function?
Best, Sebastian
Sascha Trostorff, 2019/11/14 22:45
Dear Sebastian,
yes, I think this would be the best way to argue.
Best regards
Sascha
Hendrik Vogt, 2019/11/07 15:37, 2019/11/20 21:08
Dear all,
I have a few comments on Lemma 4.2.3(a) that arose from trying to better understand why the assertion is true. First of all, for a Lipschitz continuous mapping F:L2,ν(R;H0)→L2,ν(R;H1) to be causal, it suffices to check F(f)=F(g) on (−∞,a] for simple functionsf,g∈S(R;H0) with f=g on (−∞,a]. This can be seen by the approximation argument given in the first 4 lines of the proof of 4.2.3(a).
Next I make the following observation: if c≥0, f∈S(R;H0) and g∈⋂ν≥μL2,ν(R;H1) are such that f=0 on (−∞,a] and ‖g‖2,ν≤c‖f‖2,ν for all ν≥μ, then g=0 on (−∞,a]. Indeed,
∫R‖g(t)‖2e2ν(a−t)dt≤c2∫∞a‖f(t)‖2e2ν(a−t)dt→0(ν→∞).
Since e2ν(a−t)↑∞ as ν→∞ for all t<a, it follows from the monotone convergence theorem that ∫a−∞‖g(t)‖2⋅∞dt=0, which implies the claim.
Finally, if f,g∈S(R;H0) are such that f=g on (−∞,a], then f−g=0 on (−∞,a], and ‖Fν(f)−Fν(g)‖L2,η≤L‖f−g‖L2,η for all η≥μ, with the uniform Lipschitz constant L, and by the previous paragraph it follows that Fν(f)−Fν(g)=0 on (−∞,a]. This provides a different (maybe less technical) proof of Lemma 4.2.3(a).
Now I hope that I didn't make any mistakes in my argument above!
Best wishes, Hendrik
Sascha Trostorff, 2019/11/07 16:11
Dear Hendrik,
thanks a lot for this proof. As far as I can see, you did no mistake and indeed your proof is more structured than ours.
Best regards
Sascha
Fabian Gabel, 2019/11/17 23:35
Dear Hendrik,
I like your proof very much. Is there a little typo in the paragraph starting with “Next I make the following observation:”? If I'm not mistaken, it should read g∈⋂ν≥μL2,ν(R;H1).
Best wishes,
Fabian
Hendrik Vogt, 2019/11/20 21:09
Dear Fabian,
oh yes, thanks – I've corrected the typo.
Best wishes, Hendrik
discussion/lecture_04.txt · Last modified: 2019/10/21 16:59 by matcs
Discussion on Lecture 04
Dear virtual lecturers,
I've got a question concerning the proof of Lemma 4.2.3(
c). It appears that you use the causality of Fν, but I don't really see why you use it. My argument would look as follows: The set of simple functions with compact support(!) is dense in L2,ν(R;H0)∩L2,η(R;H0), i.e., for all f∈L2,ν(R;H0)∩L2,η(R;H0) there exists a sequence (fn) of simple functions with compact support such that fn→f both in L2,ν(R;H0) and in L2,η(R;H0). This is what you show in the first 5 lines of the proof of Lemma 4.2.3(c). But from this the assertion is clear since Fν and Fη are continuous, and they coincide (by definition) on the simple functions with compact support.Am I mistaken?
Best wishes, Hendrik
Dear Hendrik,
yes, you are right. Thanks for pointing out.
Best regards
Sascha
Dear ISEM-team,
in regard to Theorem 4.2.1(d) I am tempted to say that the solution operator that maps f∈L2,ν to the solution vν,f is causal. But we only defined 'causal' for Lipschitz continuous mappings. Is the above solution operator causal? Do we have continuous dependence of the solution on the data f at all? (sorry for the naive questions)
I furthermore have a little remark on the proof of Theorem 4.3.3:
p.46, l.-7: I think it should read “by continuity of f, u0 and u.
Best wishes, Fabian
Dear Fabian,
thanks for your question. Yes, the solution operator in Theorem 4.2.1 (d) is Lipschitz-continuous, but we forgot to work that out. The argumentaion is quite simple. Choosing ν>L and uν,vν to solutions for the source terms f,g, we obtain ‖uν−vν‖=‖∂−1t,ν(Fν(uν)−Fν(vν))‖+‖∂−1t,ν(f−g)‖≤Lν‖uν−vν‖+1ν‖f−g‖ and thus, ‖uν−vν‖≤(1−L/ν)−11ν‖f−g‖.
Also thanks for spotting the missing “continuity” on p. 46.
Best regards
Sascha
Dear ISEM-team,
working through the lecture notes, I gathered the following remarks
Remark on the Proof of Theorem 4.1.1
It may be a matter of taste but reading the proof the first few times I got confused by the way the letter ψ was used:
p.39, l.3: scalar functions in C∞c(R),
p.39, l.9: H-valued functions in dom(∂t,ν).
Questions and Remarks on the Proof of Theorem 4.2.4
Usage of the phrase “As such” (p.43, .-13). I think you want to use it as a transitional phrase. Some people think that this is grammatically incorrect https://lawprofessors.typepad.com/legalwriting/2006/04/incorrect_use_o.html. What do you think?
p.44, l.3: “Furthermore, v is continuous …. As such, we obtain”. As you write it, “as such” suggests, that the equality in l.5 follows from the continuity of v, but I think that it is a consequence of the continuity of f, F and P: If I'm not mistaken, we approximate v∈dom(∂t,ν) by vn∈S(R;Kn) which leads to v=∂−1t,νFν(v)=IνFν(v)=limn→∞IνF(vn)=limn→∞∫⋅−∞1[0,δ′)(τ)f(τ,P(vn(τ)))dτ=… Does this make sense?
p.44, l.-14: “is the unique solution point of the equation of”. Do we want to use Theorem 4.2.1(b) here? Then, I would suggest to write: “is the unique fixed point of the equation w=…”
Best wishes, Fabian
Dear Fabian,
thank you very mich for your comments.
On page 39, you are right, we should be more consistent.
Concerning the phrase `as such', I don't think we are in the position to argue with a law professor.
The continuity of v is a consequence of the Sobolev embedding theorem: v∈dom(∂t,ν). The continuous differentiability of v then follows from the continuity of the integrand on the right-hand side f(⋅,Pv(⋅)).
Concerning your remark on p44, l -14: This is a misprint. We meant unique solution. The problem with your phrase is that an equation does not really have a fixed point.
Hope this helps (and I also hope I have addressed all your question correctly),
cheers,
MM
Thank you Marcus for your reply and the clarifications.
As for my calculations to justify the first equality on p.44, l.5: Did these calculations make sense or were they unnecessary? We only defined F on S(R,Kn), therefore I tried to carry out an approximation which relies on the continuity of f, F and P.
In my opinion, the text suggests that the series of equations is a mere consequence of the continuity of v. As you pointed out, this is not the case.
Best wishes, Fabian
Dear Fabian,
no, the computation has to be done, but as it is quite standard, we did not write it down. However, by the approximation the formula just holds almost everywhere, and then continuity gives the equality for each t. I hope this answers your question.
Best regards
Sascha
Yes, this answers my question. Thank you very much.
Best wishes, Fabian
Dear all,
here are some more comments from Darmstadt that are not already posted at an existing suitable position in this forum ;)
But first of all, let me say in general that we had sometimes the feeling that arguments lack help in the relevant details whereas obvious calculations are carried out with more heart. You'll probably see what we mean by this in some of the following points.
Proof of Prop. 4.1.1, second last displayed formula: In the middle term the classical derivative appears. After the displayed formula you claim that you can extend this identity to dom(∂t,ν) on using that you showed it on a core. This extension works for the left-hand and right-hand sides of the identity, which is also the relevant part, however not for the whole line as claimed.
Corollary 4.1.3: The subscript H at the norm is missing.
Definition of uniformly Lipschitz continuous: First of all, we didn't like the formulation “F considered in …”. You mean that we should consider the mapping as a relation, correct? Albeit you have introduced relations in general in Lecture two, this point of view seems not so helpful here since for example you have not introduced closability in this case. So we recommend just to consider F as a mapping between subsets of metric spaces and do the Lipschitz extension in that context.
Moreover, we wondered why the definition is done in such a seemingly complicated why. By definition of Bochner measurability, the definition turns out to work quite well but on the first sight it seems unnatural. Wouldn't it be a better definition to just say that a uniformly Lipschitz continuous function is a family of Lipschitz mappings Fν:L2,ν→L2ν for ν≥μ that are consistent and satisfy supν≥μ‖Fν‖Lip? Then, starting from simple functions could be a Lemma on how to construct uniformly Lipschitz continuous functions.
Proof of Theorem 4.2.1 first part: In the third line of the proof you say that the contraction property follows from Lemma 3.2.1. This is only half the truth. First, the relevant norm bound for ∂−1t,ν is calculated in the definition after that Lemma. Second, this is the point in the proof where the assumption ν>L is used, so you should point that out.
It was not clear to us what line 6 should tell us and the formulation “can be proved in Exercise” is also a bit strange.
In the second last line you use the term “strict contraction”, it seems like there is no difference to the term “contraction” used before?
Remark 4.2.2 We where confused by the change of rolls between ν and η in the two cases. Maybe its better understandable if their ordering is fixed before entering the cases?
Lemma 4.2.3:
1) In assertion (b) you have the additional constraint that ν>0. This comes from the fact that only then ∂−1t,ν is causal. In the proof you only write that this follows directly from (a)…
2) In assertion © and at the beginning of the proof a space H appears. Probably this should be H0?
3) Above the first displayed formula you have these arguments in which spaces 1(−∞,a]⋅x is and in which spaces fn−gn converges to f−g. First, you should indicate the arguments for the convergence of fn−gn (they have different roles, which results in a function with support near +∞, which decides with embedding from Remark 4.2.2 to use). While doing this, it turns out that the constraint η≥0 is not needed for this argument, in contrast to what is indicated in the text. Then you tell that with the correct combined constraints for η the following calculation follows, but it is not indicated where the previously shown assertions are used. This happens at the first limit, which is not at all explicitly explained. The arguments to justify this limit are straightforward, but you should at least tell the reader to use Hölder's inequality with the previous assertions to point into the right direction to do the standard arguments on its own.
4) In the last estimate of the first displayed formula you use the (to us) long forgotten assumption (:P) that f and g coincide on (−∞,a] together with the embeddings from the Remark. A hint that this assumption is used here would be helpful.
5) After the first displayed formula you let η→∞ which amounts to show that the integral with which one has started vanishes However, this doesn't show that Fν(f)=Fν(g). We suggest to do the calculation with 1A for all measurable subsets A⊆(−∞,a] instead of only with the maximal indicator function to get the claim.
Proof of Theorem 4.2.1 second part: First: To go from the first to the second line in the first displayed formula you use again that ∂−1t,ν is causal (cf. 4.2.3 (b)). Second: You justify the last displayed formula by “again using Lemma 4.2.3”. This is misleading. Indeed, the formula follows from Lemma 4.2.3, but in contrast to before, one has to use part © of that Lemma. But it is also possible to use part (b) before past © (if not using the previously shown part (b) of that Theorem) which could lead to confusion.
Proof of Theorem 4.2.4: In the last displayed formula on p. 43, the δ should be δ′.
Lemma 4.3.1: In addition to what Jürgen has already mentioned: We suggest to distinguish between the equivalence part and the “Moreover”-part. In the first we suggest to introduce the parameter μ∈R as Jürgen said, but one should also quantify ν in the “Moreover”-part and the quantification should be over R, though in the proof in the last line the supremum should be taken over ν≥μ.
Best, Sebastian
Dear Sebastian,
thanks a lot for your remarks and suggestions. Let me comment on them:
Proof of Prop. 4.1.1, second last displayed formula: In the middle term the classical derivative appears. After the displayed formula you claim that you can extend this identity to dom(∂t,ν) on using that you showed it on a core. This extension works for the left-hand and right-hand sides of the identity, which is also the relevant part, however not for the whole line as claimed.
Of course you are right and just the term on the left- and on the right-hand side can be extended to dom(∂t,ν).
Definition of uniformly Lipschitz continuous: First of all, we didn't like the formulation “F considered in …”. You mean that we should consider the mapping as a relation, correct? Albeit you have introduced relations in general in Lecture two, this point of view seems not so helpful here since for example you have not introduced closability in this case. So we recommend just to consider F as a mapping between subsets of metric spaces and do the Lipschitz extension in that context.
I think this is a matter of taste. In general, mappings (linear or non-linear) are nothing but relations, so I do not see why our formulation is worse than yours. And yes, we did not consider closabilty in this case but we thought that taking closures of Lipschitz-continuous functions is quite standard (although, it is maybe not standard to note that the continuation is actually nothing but taking the closure).
Moreover, we wondered why the definition is done in such a seemingly complicated why. By definition of Bochner measurability, the definition turns out to work quite well but on the first sight it seems unnatural. Wouldn't it be a better definition to just say that a uniformly Lipschitz continuous function is a family of Lipschitz mappings Fν:L2,ν→L2ν for ν≥μ that are consistent and satisfy supν≥μ∥Fν∥Lip? Then, starting from simple functions could be a Lemma on how to construct uniformly Lipschitz continuous functions.
Again I would say that this is a matter of taste. However, let me emphasise that it is important to define “consistent” in your definition properly. So, for instance, it is not enough to say the the functions Fν coincide on a subset D dense in each L2,ν, since then causality would not follow.
It was not clear to us what line 6 should tell us and the formulation “can be proved in Exercise” is also a bit strange.
Well, okay. Maybe one should write that differentiability follows by Exercise 4.1 and the fact that Fν(uν) is continuous.
In the second last line you use the term “strict contraction”, it seems like there is no difference to the term “contraction” used before?#
Of course not. That was just sloppy writing. Thanks for pointing out.
Remark 4.2.2 We where confused by the change of rolls between ν and η in the two cases. Maybe its better understandable if their ordering is fixed before entering the cases?
Yes, thank you!
Lemma 4.2.3:
1) That is one way to see it. The other way would be to say that ∂−1t,νF is only a uniformly Lipschitz-continuous mapping, if ν>0.
2) Yes, this should be H0, thank you.
3) I agree, that the arguments are maybe too short. We apologise for this. And thanks for pointing out, that η≥0 is not needed for the convergence of fn−gn.
4) You are right. In the final notes we will follow the proof suggested by Hendrik, which is more structured.
5) Correct, thanks!
Proof of Theorem 4.2.1 second part: First: To go from the first to the second line in the first displayed formula you use again that ∂−1t,ν is causal (cf. 4.2.3 (b)).
Correct!
Second: You justify the last displayed formula by “again using Lemma 4.2.3”. This is misleading. Indeed, the formula follows from Lemma 4.2.3, but in contrast to before, one has to use part © of that Lemma. But it is also possible to use part (b) before past © (if not using the previously shown part (b) of that Theorem) which could lead to confusion
Indeed, we thought to use part (b) and © of the Lemma. We will indicate this in the final notes, thank you!
Proof of Theorem 4.2.4: In the last displayed formula on p. 43, the δ should be δ′.
Thanks!
Lemma 4.3.1: In addition to what Jürgen has already mentioned: We suggest to distinguish between the equivalence part and the “Moreover”-part. In the first we suggest to introduce the parameter μ∈R as Jürgen said, but one should also quantify ν in the “Moreover”-part and the quantification should be over R, though in the proof in the last line the supremum should be taken over ν≥μ
You are right, we will rewrite it.
Best regards
Sascha
Dear Sascha,
concerning consistency: My definition is what you check in Theorem 5.3.5 in the next lecture.
Best, Sebastian
Dear Sebastian,
you are right, however there might still be examples, where you find a nice expression for F on simple functions and not on the L2 spaces (like for the Bochner integral or for the Fourier transformation), where it seems to be more natural to define the function on that set and consider the extensions of this mapping. Anyway, it does not make any difference in the arguments need to be applied.
Best regards
Sascha
Dear Sascha,
I agree that there are cases in which the simple functions are a better starting point, my ´question´ was more or less the suggestion to start with a concise definition and put this special construction in a lemma then directly making it the definition, but that´s a matter of taste and as I said, in your usage the definition doesn´t seem so unnatural afterwards.
Best, Sebastian
Dear all,
there is a mistake in Exercise 4.7 (b). One also has to require that for each a∈R and f,g∈D=dom(F) with f=g on (−∞,a] it follows that F(f)=F(g) on (−∞,a] (so, F is causal on D).
Best regards
Sascha
Dear virtual lectures,
just before Lemma 4.3.4 you define u(.):R∋t↦(R≤0∋θ↦u(t+θ)∈H), which can be can be really confusing. Especially because you probably wanted it to be the other way round. I think it should be a family in θ∈R≤0 of functions in t∈R, but as it is written I read it as a family in t of functions in θ.
I would recommand to avoid this completely, by defining uθ:{R→H,t↦u(t+θ), for θ∈R≤0.
Some minor comments
Dear virtual lecturers,
Proposition 4.1.1: The final statement is (minimally) confusing, because g doesn't appear in (i). The remedy would be to start out with f,g∈L2,ν, and (i) should be f∈dom(∂t,ν), ∂t,νf=g.
Theorem 4.1.2: `representer' should be `representative'. And I do not at all understand the last line of the proof.
Line below Corollary 4.1.3: Postpone a proof would mean to do it later (I think).
Theorem 4.1.2(a): I simply do not know what `continuously differentiable at' a point means. As you do not prove this part, it does not become clear from the proof.
Lemma 4.2.3(c), misprint: The `μ' in the last space should be `η'.
Lemma 4.3.1: `uniformly Lipschitz continuous' is defined in connection with a μ∈R; see the definition on p.40. But here no μ is indicated.
4.5 Comments, line 1: you probably mean ODE (or not?).
And a comment on style: `obtain that' is verboten.
Best wishes, Jürgen
Dear virtual lecturers,
in the proof of Proposition 4.1.1 you quote Prop. 3.2.4 and Cor. 3.2.6, and in both of them one has the hypothesis ν≠0, whereas Prop. 4.1.1 is stated for all ν∈R. Okay, the last statement in Cor. 3.2.7 just says that Cor. 3.2.6 also holds for ν=0. Probably one should mention somewhere that the definition of ∂t,0 implies that Prop. 3.2.4 is also true for ν=0.
Best wishes, Jürgen
Dear Jürgen,
your are completely right! Thanks!
Best regards,
moppi
A further comment on the references: I think you mean DH (tensorizations of C∞c(R) with H) instead of C∞c(R;H). The latter (which is the version in the notes) would require to reference Prop. 3.2.3 instead as far as I see.
Best, Sebastian
Dear Jürgen,
thank you for your remarks!
Your comment on Prop 4.1.1 is very helpful. Concerning Theorem 4.1.2; the word representer does exist and has also been used in this context. It means the same as representative, anyway. I'll ask the native speakers in Glasgow, whether either of these words is fine, though. I will do the same for `postponed'.
The statement in Theorem 4.1.2(a) should rather read `continuously differentiable in a neighbourhood of a'.
Your are right concerning Lemma 4.3.1; we have not been precise there. One can choose any μ.
We actually mean PDE. I have not seen many ODE approaches using Lp-spaces. We want to stress that the developed ODE theory has similarities to PDE theory, where you generalise differentiability and/or solution concepts. A posteriori one tries to find out, whether one can show that the PDE is satisfied point wise or whether the derivative operators are actually classical derivatives. For instance in the case of Poisson's equation on a bounded domain. This may be solved by an application of Riesz-representation theorem in L2 and afterwards one shows given a suitably regular right-hand side, whether the solution is smoother so that the applied Laplace may even be applied point wise to the solution.
Also thank you for the style comment.
Best regards,
moppi
Dear Jürgen,
I have some comments and questions on some of the stylistic points you raised on the lecture.
“Theorem 4.1.2: `representer' should be `representative'. And I do not at all understand the last line of the proof.”
I had to do a double take on this; I had originally thought that the two were synonymous with one another, it turns out that they are not. It's funny that they pertain to tantamount the same entity, but representer (or representor) only applies to a living person in a particular capacity. I need to remember this point, thanks!
“Line below Corollary 4.1.3: Postpone a proof would mean to do it later (I think).”
Is your point that despite the virtual lecturers writing that the proof is 'postponed' till later, they don't, by the close of the lecture, provide a proof in writing themselves? I think that the usage of 'postpone' still works in this context because, as we have seen with the comments, suggestions and exercise attempts through the previous sections, the ISem is a kind of “living” lecture series, where the insights, feedback and exercise attempts from all participants go on to form the material of the series.
“And a comment on style: `obtain that' is “And a comment on style: `obtain that' is verboten.”verboten.”
Can you clarify as to why this turn of phrase is forbidden?
Cheers,
Michael
Dear Michael,
I understand that I'm on dangerous ground here, not being a native speaker in contrast to you.
Concerning `representer', I was under the impression that the word doesn't even exist, and my suspicion was confirmed by its non-existence at LEO. Meanwhile I found that it does exist, but anyway, I had never seen it in the present context.
Concerning `postpone', I still would expect that something that is postponed will be treated in the text by the authors, but OK, it may be disputable. But maybe I can ask you whether the formulation `The proof is delegated to Exercise 4.2' would be acceptable for expressing that it will be posed as an exercise.
Concerning `obtain that', let me say that in mathematical German one can formulate “Von Theorem .. erhalten wir, dass v(0)=0 ist.” Translating word by word one obtains `By Theorem .. we obtain that v(0)=0.', and I have been told that `obtain' can only be used in connection with nouns. One can obtain things or results, for instance. Correct?
Best wishes, Jürgen
Dear Jürgen,
On the representer point, you were actually correct that its usage here was not right - so you actually taught me something here! (It's funny to note that both representer and representative effectively relate to the same thing; it's just that the former applies only to persons!).
Sure, I think that within the “living” or “evolving” text perspective its usage holds up. I think that the issue with “delegate”, much like with “representer” before, is that they only apply to persons. For instance, one talks about delegating a task to another person.
I have a feeling that you might be right, again, on the usage of “obtain that”. I have found this post here which seems to elucidate the point, which until now I had completely overlooked: https://english.stackexchange.com/questions/92067/how-do-i-use-the-word-obtain-together-with-mathematical-formulas
Cheers again for pointing these things out - they can only help to better one's style in presenting mathematics.
Best wishes,
Michael
Dear Michael,
and isn't it funny who asked that question on stackexchange?
Best wishes, Hendrik
Dear Hendrik,
Funny indeed!
Cheers,
Michael
Dear Michael,
incidentally, I also found a dangling participle on p. 42, line -4.
Best wishes, Jürgen
Hi Jürgen,
Do you mean the sentence: “Hence, again using Lemma 4.2.3, it follows that”? If so, well spotted!
Cheers,
Michael
Dear Jürgen,
“And I do not at all understand the last line of the proof.”
I agree that this is hardly understandable from the written text. The given calculation shows the norm estimate without using anything else and for the continuity one has to redo the calculation using f(t)−f(tn), which amounts to restrict the range of integration in the calculation. Clearly, the shown calculation then gives a dominating function (1(−∞,t+ε)‖g‖) and we can use the dominated convergence theorem to conclude continuity.
Best, Sebastian
Dear Sebastian, dear Jürgen,
the reference to dominated convergence is made to see that t↦∫t−∞g(s)ds is continuous for g∈L2,ν. However, I do not see why you have to redo the calculation as Sebastian indicated.
Best regards
Sascha
Dear Sascha,
OK; now I understand that you use the estimate to see that g is integrable over intervals (−∞,a] for all a∈R, and then you also use the definition of f. (This is not reflected by the sequence of statements, in particular, because the estimate starts with the estimate for ‖f(t)‖). And why do you need dominated convergence for the desired norm estimate? (The given estimate is quite sufficient for the assertion.)
Best wishes, Jürgen
Dear Jürgen,
of course you don’t need dominated convergence for the estimate but just for the continuity. The formulation in the text is misleading, sorry for that.
Best regards
Sascha
Dear Sascha,
you are right, one doesn't have to redo the whole calculation. But at least the first step. Maybe it would be better to first show the integrability of ‖g‖ over intervals with the respective estimate and then use this to conclude the estimate on the one hand and secoundly perform the continuity argument using dominated convergence and the previously established dominating function?
Best, Sebastian
Dear Sebastian,
yes, I think this would be the best way to argue.
Best regards
Sascha
Dear all,
I have a few comments on Lemma 4.2.3(a) that arose from trying to better understand why the assertion is true. First of all, for a Lipschitz continuous mapping F:L2,ν(R;H0)→L2,ν(R;H1) to be causal, it suffices to check F(f)=F(g) on (−∞,a] for simple functions f,g∈S(R;H0) with f=g on (−∞,a]. This can be seen by the approximation argument given in the first 4 lines of the proof of 4.2.3(a).
Next I make the following observation: if c≥0, f∈S(R;H0) and g∈⋂ν≥μL2,ν(R;H1) are such that f=0 on (−∞,a] and ‖g‖2,ν≤c‖f‖2,ν for all ν≥μ, then g=0 on (−∞,a]. Indeed, ∫R‖g(t)‖2e2ν(a−t)dt≤c2∫∞a‖f(t)‖2e2ν(a−t)dt→0(ν→∞). Since e2ν(a−t)↑∞ as ν→∞ for all t<a, it follows from the monotone convergence theorem that ∫a−∞‖g(t)‖2⋅∞dt=0, which implies the claim.
Finally, if f,g∈S(R;H0) are such that f=g on (−∞,a], then f−g=0 on (−∞,a], and ‖Fν(f)−Fν(g)‖L2,η≤L‖f−g‖L2,η for all η≥μ, with the uniform Lipschitz constant L, and by the previous paragraph it follows that Fν(f)−Fν(g)=0 on (−∞,a]. This provides a different (maybe less technical) proof of Lemma 4.2.3(a).
Now I hope that I didn't make any mistakes in my argument above!
Best wishes, Hendrik
Dear Hendrik,
thanks a lot for this proof. As far as I can see, you did no mistake and indeed your proof is more structured than ours.
Best regards
Sascha
Dear Hendrik,
I like your proof very much. Is there a little typo in the paragraph starting with “Next I make the following observation:”? If I'm not mistaken, it should read g∈⋂ν≥μL2,ν(R;H1).
Best wishes, Fabian
Dear Fabian,
oh yes, thanks – I've corrected the typo.
Best wishes, Hendrik