I would like to share some comments from our meeting in Darmstadt.
In our opinion it is reasonable to mention Lemma 2.2.7 in the proof of Proposition 6.1.1
In Remark 6.1.3 it is stated that H1(Ω)⊆H(div,Ω)∩H(curl,Ω). It should rather be H1(Ω)3 instead of H1(Ω).
In our opinion it is reasonable to mention Corollary 2.2.6 in the proof of Proposition 6.1.5
On page 69 in the first line it is talked about the fact that for well behaved F there is a solution. At this point it is omitted that there are conditions for M and A too. Well, this is quite obvious. But it is also obvious that F needs to be well in a certain sense. The main point here should be that in the Theorem 6.2.1 the conditions for F are relatively strong.
In Propostioon 6.2.2 b) we suggest to use A instead of a for the notation of a linear operator for consistency reasons.
In the proof of Proposition 6.2.2 in the third indented equation from below in the first term it is omitted to mark the first norm as H0-norm. Furthermore in the second term after the inequality sign the H0 have to be before the line to mark the scalar product as a H0-scalar product.
In the same proof the argument in the two sentences after the second indented equation from below (the argument why a−1∈L(H0)) is confusing. Better way: …that a−1 is bounded. Lemma 2.1.3. now implies that dom(a−1)⊆H0 is closed. Since ran(a)=dom(a−1) we conclude that dom(a−1) is also dense and i.e. equal to H0. Thus a−1∈L(H0).
The arguments in Theorem 6.2.3. and Theorem 6.2.5. is almost the same. But in the latter Theorem the argumentation is more precise and detailed. It is more reasonable to do it the other way around so that the first Theorem is more detailed.
In the proof of Theorem 6.2.1. in the first two lines the condition ν≤ν0 is redundant.
In the same proof at page 77 in the first line it should be “for almost all t ∈R” instead of “for all t ∈R.
At page 77 after the third indented equation it is said that the right hand side is the closure of the relation (inverse of something). But we identify the right-hand side as the inverse of the closure of (something). This is confusing.
Best regards
Team Darmstadt
in this post represented by Markus
Marcus Waurick, 2019/12/04 07:34
Dear Darmstadt Team, dear Markus,
thank you very much for your observation and comments.
1. Yes, this clarifies something.
2. Yes, thanks.
3. Yes, thanks.
4. This is an interesting view on Picard's Theorem. The comment on `well-behaved' is that F is in the domain of ∂t in order to get a solution in the operator sum without the need for the closure bar (see also Hendrik's comment). Later we shall see that there are also unique solutions for considerably less regular or less well-behaved F. This was the point of our motivating part.
5. I agree that it might make sense to use a capital letter for this. We can however not use A as this is almost always used to denote the potentially unbounded skew-selfadjoint spatial operator in evolutionary equations. In applications to the heat equation for instance the coefficient is rather a and not A. This was the reason for us to use a.
6. Yes, thanks.
7. This is the argument. We have been a bit short on this. Thanks.
8. The reason that we chose to present the proof of Theorem 6.2.5 in more details is the following: In 6.2.3 we don't need self-adjointness of the coefficient. Hence, the positive definiteness follows easily from 6.2.2. In 6.2.5, however, we need to argue, why self-adjointness is needed for T.
9. Yes, thanks.
10. Yes, thanks.
11. I don't really see your point. Could you clarify this a bit more, please? In Lecture 2, we have shown that closure and inversion are interchangeable processes for relations.
Best wishes,
MM
Markus Borkowski, 2019/12/04 15:49
Dear Marcus,
your replies for point 8 and 11 clarify the confusion.
Thanks
Markus B
Sebastian Bechtel, 2019/12/04 23:27
I disagree ;)
Concerning 8, I would argue that the part using self-adjointness is quite small compared to the whole calculation that coincides with the previous case where it is stated as being totally trivial. I guess changing the order of examples is no option for you? Otherwise, I would suggest to perform the calculation in the heat case properly and in the wave case just explain the new ingredients (or perform the calculation a second time, in my opinion it really doesn't take so much space and hence it should be apropriate in the context of lecture notes).
For 11, do you have a reference? I do agree that this interchanging holds and it is not so hard to see this, but I couldn't find a reference in Lecture 2. Could you tell where this is explicitly stated?
Best, Sebastian
Marcus Waurick, 2019/12/05 09:09, 2019/12/05 09:09
Dear Sebastian,
thank you for your comments.
Changing the order would not be in the interest of the didactic of the whole manuscript. In Lecture 1, we started out with the heat equation in order to have a natural entry point from the perspective of C0-semigroups. In this situation the difference between evolutionary equations and evolution equations becomes (in my opinion) most apparent.
It is always a question how many details one provides for the proofs of certain statements. I agree that we can extend the part with heat equation a bit more. In any case, I hope the proof of the well-posedness statement is understandable as is (even though it might a bit short).
Concerning 11, I'm sorry I don't have an explicit reference. It is also possible to argue that we have omitted the statement asked for (without deliberation, though). However, if one wonders, whether closure and inversion are interchangeable, one might have a look at Proposition 2.1.1 (a) and its proof. The argument presented in the proof eventually implies the statement.
I hope my explanations above are of some help.
Best regards,
MM
Sebastian Bechtel, 2019/12/05 11:16
Dear Moppi,
your comment concerning order is convincing!
Personally, I can live with the version as it is, because I take out a piece of paper and do the calculation if a step is not immediate to me, and the second fact about the relations was known to me. I just argue from the perspective of someone who might get insecure because a step that is stated as obvious is not immediate to them. Maybe it would already help to change the wording in such a way that it is clear that there is a short calculation to do for which most people will need pen and paper… ?
Best, Sebastian
Marcus Waurick, 2019/12/06 10:52
Dear Sebastian,
thank you for your comments. Yes, I agree that it would have helped to just note a sentence or two to make things more transparent. Also, I think it doesn't hurt to include a comment in Lecture 2 after Proposition 2.1.1 that inversion and closure are interchangeable.
It is very important for us to have these kind of discussions in the forum. We shall include some more details in the very final version particularly concerning the application of Picard's Theorem.
Best regards,
moppi
Hendrik Vogt, 2019/11/28 09:19
Dear all,
I've got another comment on Picard's theorem 6.2.1. As far as I understand the theorem, it implies that the equation ∂t,νM(∂t,ν)U+AU=F has a unique solution U∈dom(∂t,ν)∩dom(A) if F∈dom(∂t,ν). My point is that in the equation, the term M(∂t,ν)U is in the domain of ∂t,ν (and the term U is in the domain of A), so that no closures are needed if F∈dom(∂t,ν). Of course, one has to note that M(∂t,ν) maps dom(∂t,ν) to itself.
This might seem like a trivial observation, but I didn't find it mentioned in the lecture, and it is used later on, e.g. at the top of page 72. At that place, an explanation is given for the particular case of the heat equation. Thus I thought it's worth pointing out that an analogous observation holds generally.
Best wishes, Hendrik
Marcus Waurick, 2019/11/28 15:49
Dear Hendrik,
I can only emphasise that this is indeed a very valuable comment. We should have pointed out this more. We will have a occasion to use this observation later on, also in this abstract situation.
Best regards,
MM
Hendrik Vogt, 2019/11/21 10:51
Dear all,
I'd like to comment on the proof of Picard's theorem 6.2.1. It seems to me that the proof can be simplified substantially; in particular, the argument for closability in the first paragraph is not needed at all, as far as I see.
It appears to me that the core of the proof is the bottom 10 lines on page 76 (plus the 7 lines at the top of page 77). Based on the facts shown there I'd argue as follows. The operator N(im+ν) on L2(R;H) is bounded and injective, so its inverse is a closed multiplication operator that I'd like to denote by B(im+ν). More explicitly, B(im+ν)f=[t↦B(it+ν)f(t)]=:gf for all f∈L2(R;H) that satisfy f(t)∈dom(A) for a.e. t∈R and gf∈L2(R;H). By the definition of B, B(im+ν) is an extension of (im+ν)M(im+ν)+A, so the latter operator is closable.
Now it remains to show that B(im+ν) is the closure of (im+ν)M(im+ν)+A, or expressed differently, that dom((im+ν)M(im+ν))∩dom(A) is a core for B(im+ν). For the proof of this let f∈dom(B(im+ν)). Then
fn:=1[−n,n]f∈dom(B(im+ν)),B(im+ν)fn=1[−n,n]B(im+ν)f→B(im+ν)f
in L2(R;H) as n→∞. Moreover, fn∈dom((im+ν)M(im+ν)) since fn has compact support, and this also implies that fn∈dom(B(im+ν)−(im+ν)M(im+ν))⊆dom(A). Thus the core property is proved.
Now I'm curious to learn the virtual lectures' opinion about this Am I missing some argument?
Best wishes, Hendrik
P.S.: I have to say that I'm not quite happy with the notation (im+ν)M(im+ν)+A since it is not obvious from the notation that this is a sum of two operators and not the operator of multiplication by [t↦(it+ν)M(it+ν)+A].
Sascha Trostorff, 2019/11/21 11:22
Dear Hendrik,
first of all let me thank you for reading our manuscript so carefully. I am really happy that you support us by clarifying arguments and providing more elegant proofs. Again, your are completely right with your observations above. For some reason (I don't know why) we have tried to avoid multiplication operators arising from functions which are closed operator-valued, like the operator B(im+ν) in your post. But you convinced me that we should work with it.
Concerning the notation: Yes, I agree, that this is a problem and indeed, some year ago it took me some time to notice the subtle difference between these two objects. That was the reason for the remark in the proof of Theorem 6.2.1 in the middle of p. 77.
Best regards
Sascha
Hendrik Vogt, 2019/11/21 12:36
Dear Sascha,
thanks for your prompt answer! Now I have another question: it is crucial for the proof that N is a material law, but where is it used that M is a material law? In other words, could it be that the assumptions on M can be relaxed?
I would imagine the following more general theorem: if K:dom(K)⊆C→L(H) is holomorphic on some right half-plane, and Re⟨ϕ,K(z)ϕ⟩≥c‖ϕ‖2H for all ϕ∈H and all z from that half-plane, for some c>0, then one obtains the same assertions for the operator K(∂t,ν)+A as for ∂t,νM(∂t,ν)+A. I didn't have the time yet to see if the proof that I indicated above goes through as is.
yes, indeed you can relax the assumptions on M. Indeed, you just need analyticity on some half-plane, the invertibility of zM(z)+A and a uniform estimate for the operator norms of the inverses on some right half-plane (I did this in my habilitation thesis https://arxiv.org/abs/1707.00429). However, I am not aware of any example from mathematical physics, where you really need these weaker assumptions.
Best regards
Sascha
Hendrik Vogt, 2019/11/21 14:11
Dear Sascha,
thank you! As you might expect, I'm not thinking from the perspective of applications. Don't understand me wrong: I find it exciting how many applications the theory has! What I mean is that with the assumptions I suggested, the notation in the proof becomes simpler, and I also find it easier to understand the theory if one works with the minimal set of assumptions. Let me point out that I can well imagine that for reasons of presentation it might not be so easy to follow my suggestion.
Best wishes, Hendrik
Alexander Dobrick, 2019/11/21 13:57
Dear Hendrik,
as far as I understand your argument above, I agree that the generalization of the theorem you indicated in the second post should hold.
Best wishes, Alex
discussion/lecture_06.txt · Last modified: 2019/10/21 17:00 by matcs
Discussion on Lecture 06
Dear virtual lecturers,
I would like to share some comments from our meeting in Darmstadt.
Best regards
Team Darmstadt
in this post represented by Markus
Dear Darmstadt Team, dear Markus,
thank you very much for your observation and comments.
1. Yes, this clarifies something.
2. Yes, thanks.
3. Yes, thanks.
4. This is an interesting view on Picard's Theorem. The comment on `well-behaved' is that F is in the domain of ∂t in order to get a solution in the operator sum without the need for the closure bar (see also Hendrik's comment). Later we shall see that there are also unique solutions for considerably less regular or less well-behaved F. This was the point of our motivating part.
5. I agree that it might make sense to use a capital letter for this. We can however not use A as this is almost always used to denote the potentially unbounded skew-selfadjoint spatial operator in evolutionary equations. In applications to the heat equation for instance the coefficient is rather a and not A. This was the reason for us to use a.
6. Yes, thanks.
7. This is the argument. We have been a bit short on this. Thanks.
8. The reason that we chose to present the proof of Theorem 6.2.5 in more details is the following: In 6.2.3 we don't need self-adjointness of the coefficient. Hence, the positive definiteness follows easily from 6.2.2. In 6.2.5, however, we need to argue, why self-adjointness is needed for T.
9. Yes, thanks.
10. Yes, thanks.
11. I don't really see your point. Could you clarify this a bit more, please? In Lecture 2, we have shown that closure and inversion are interchangeable processes for relations.
Best wishes,
MM
Dear Marcus,
your replies for point 8 and 11 clarify the confusion.
Thanks
Markus B
I disagree ;)
Concerning 8, I would argue that the part using self-adjointness is quite small compared to the whole calculation that coincides with the previous case where it is stated as being totally trivial. I guess changing the order of examples is no option for you? Otherwise, I would suggest to perform the calculation in the heat case properly and in the wave case just explain the new ingredients (or perform the calculation a second time, in my opinion it really doesn't take so much space and hence it should be apropriate in the context of lecture notes).
For 11, do you have a reference? I do agree that this interchanging holds and it is not so hard to see this, but I couldn't find a reference in Lecture 2. Could you tell where this is explicitly stated?
Best, Sebastian
Dear Sebastian,
thank you for your comments.
Changing the order would not be in the interest of the didactic of the whole manuscript. In Lecture 1, we started out with the heat equation in order to have a natural entry point from the perspective of C0-semigroups. In this situation the difference between evolutionary equations and evolution equations becomes (in my opinion) most apparent.
It is always a question how many details one provides for the proofs of certain statements. I agree that we can extend the part with heat equation a bit more. In any case, I hope the proof of the well-posedness statement is understandable as is (even though it might a bit short).
Concerning 11, I'm sorry I don't have an explicit reference. It is also possible to argue that we have omitted the statement asked for (without deliberation, though). However, if one wonders, whether closure and inversion are interchangeable, one might have a look at Proposition 2.1.1 (a) and its proof. The argument presented in the proof eventually implies the statement.
I hope my explanations above are of some help.
Best regards,
MM
Dear Moppi,
your comment concerning order is convincing!
Personally, I can live with the version as it is, because I take out a piece of paper and do the calculation if a step is not immediate to me, and the second fact about the relations was known to me. I just argue from the perspective of someone who might get insecure because a step that is stated as obvious is not immediate to them. Maybe it would already help to change the wording in such a way that it is clear that there is a short calculation to do for which most people will need pen and paper… ?
Best, Sebastian
Dear Sebastian,
thank you for your comments. Yes, I agree that it would have helped to just note a sentence or two to make things more transparent. Also, I think it doesn't hurt to include a comment in Lecture 2 after Proposition 2.1.1 that inversion and closure are interchangeable.
It is very important for us to have these kind of discussions in the forum. We shall include some more details in the very final version particularly concerning the application of Picard's Theorem.
Best regards,
moppi
Dear all,
I've got another comment on Picard's theorem 6.2.1. As far as I understand the theorem, it implies that the equation ∂t,νM(∂t,ν)U+AU=F has a unique solution U∈dom(∂t,ν)∩dom(A) if F∈dom(∂t,ν). My point is that in the equation, the term M(∂t,ν)U is in the domain of ∂t,ν (and the term U is in the domain of A), so that no closures are needed if F∈dom(∂t,ν). Of course, one has to note that M(∂t,ν) maps dom(∂t,ν) to itself.
This might seem like a trivial observation, but I didn't find it mentioned in the lecture, and it is used later on, e.g. at the top of page 72. At that place, an explanation is given for the particular case of the heat equation. Thus I thought it's worth pointing out that an analogous observation holds generally.
Best wishes, Hendrik
Dear Hendrik,
I can only emphasise that this is indeed a very valuable comment. We should have pointed out this more. We will have a occasion to use this observation later on, also in this abstract situation.
Best regards,
MM
Dear all,
I'd like to comment on the proof of Picard's theorem 6.2.1. It seems to me that the proof can be simplified substantially; in particular, the argument for closability in the first paragraph is not needed at all, as far as I see.
It appears to me that the core of the proof is the bottom 10 lines on page 76 (plus the 7 lines at the top of page 77). Based on the facts shown there I'd argue as follows. The operator N(im+ν) on L2(R;H) is bounded and injective, so its inverse is a closed multiplication operator that I'd like to denote by B(im+ν). More explicitly, B(im+ν)f=[t↦B(it+ν)f(t)]=:gf for all f∈L2(R;H) that satisfy f(t)∈dom(A) for a.e. t∈R and gf∈L2(R;H). By the definition of B, B(im+ν) is an extension of (im+ν)M(im+ν)+A, so the latter operator is closable.
Now it remains to show that B(im+ν) is the closure of (im+ν)M(im+ν)+A, or expressed differently, that dom((im+ν)M(im+ν))∩dom(A) is a core for B(im+ν). For the proof of this let f∈dom(B(im+ν)). Then fn:=1[−n,n]f∈dom(B(im+ν)),B(im+ν)fn=1[−n,n]B(im+ν)f→B(im+ν)f in L2(R;H) as n→∞. Moreover, fn∈dom((im+ν)M(im+ν)) since fn has compact support, and this also implies that fn∈dom(B(im+ν)−(im+ν)M(im+ν))⊆dom(A). Thus the core property is proved.
Now I'm curious to learn the virtual lectures' opinion about this
Am I missing some argument?
Best wishes, Hendrik
P.S.: I have to say that I'm not quite happy with the notation (im+ν)M(im+ν)+A since it is not obvious from the notation that this is a sum of two operators and not the operator of multiplication by [t↦(it+ν)M(it+ν)+A].
Dear Hendrik,
first of all let me thank you for reading our manuscript so carefully. I am really happy that you support us by clarifying arguments and providing more elegant proofs. Again, your are completely right with your observations above. For some reason (I don't know why) we have tried to avoid multiplication operators arising from functions which are closed operator-valued, like the operator B(im+ν) in your post. But you convinced me that we should work with it.
Concerning the notation: Yes, I agree, that this is a problem and indeed, some year ago it took me some time to notice the subtle difference between these two objects. That was the reason for the remark in the proof of Theorem 6.2.1 in the middle of p. 77.
Best regards
Sascha
Dear Sascha,
thanks for your prompt answer! Now I have another question: it is crucial for the proof that N is a material law, but where is it used that M is a material law? In other words, could it be that the assumptions on M can be relaxed?
I would imagine the following more general theorem: if K:dom(K)⊆C→L(H) is holomorphic on some right half-plane, and Re⟨ϕ,K(z)ϕ⟩≥c‖ϕ‖2H for all ϕ∈H and all z from that half-plane, for some c>0, then one obtains the same assertions for the operator K(∂t,ν)+A as for ∂t,νM(∂t,ν)+A. I didn't have the time yet to see if the proof that I indicated above goes through as is.
Best wishes, Hendrik
Dear Hendrik,
yes, indeed you can relax the assumptions on M. Indeed, you just need analyticity on some half-plane, the invertibility of zM(z)+A and a uniform estimate for the operator norms of the inverses on some right half-plane (I did this in my habilitation thesis https://arxiv.org/abs/1707.00429). However, I am not aware of any example from mathematical physics, where you really need these weaker assumptions.
Best regards
Sascha
Dear Sascha,
thank you! As you might expect, I'm not thinking from the perspective of applications. Don't understand me wrong: I find it exciting how many applications the theory has! What I mean is that with the assumptions I suggested, the notation in the proof becomes simpler, and I also find it easier to understand the theory if one works with the minimal set of assumptions. Let me point out that I can well imagine that for reasons of presentation it might not be so easy to follow my suggestion.
Best wishes, Hendrik
Dear Hendrik,
as far as I understand your argument above, I agree that the generalization of the theorem you indicated in the second post should hold.
Best wishes, Alex