p. 149, line 6: The second and third term should carry a minus sign.
It occured to me that for some participants it must be difficult to absorb the definition of H−1/2 (in Theorem 12.2.2). After all, Hilbert spaces are self-dual. Probably one should insert some explanation concerning Gelfand triples.
In the very last part of the proof of Corollary 12.2.3, we did not understand why the property div0q=divq is mentioned.
Best wishes, Jürgen
Marcus Waurick, 2020/02/02 12:49
Dear Jürgen,
thank you for your comments!
p 149, l 6: You are right, thanks!
We should mentioned the notion of Gelfand triples at least in a remark somewhere. This is true. I think we should place this remark already in Lecture 9, where we introduced for instance H−1(Ω).
In the last part of Corollary 12.2.3 we sticked too much to the definition of the adjoint: After all we have an equation of the form
⟨p,f⟩+⟨q,gradf⟩=0for some p valid for all f∈dom(grad). Thus, q∈dom(grad∗) and grad∗f=p. We have applied this to p=divq. But you are right, this was nothing we wanted to show really, as the only thing needed was to assert q∈dom(grad∗). Thanks!
Best regards,
MM
Gabriel McCracken, 2020/01/30 15:40, 2020/01/30 15:40
Dear all,
as Jürgen already pointed out most, we've only got two grammatical remarks left:
In line 1 of 12.1, “is” is double.
The first passage of 12.3 bears several problems: “We shall address other boundary value problems in the exercises” when talking about inhomogeneous b.v.p's before conveys the impression that we address homogeneous b.v.p's in the exercises. What you probably mean is b.v.p's other than scalar wave type equations with Neumann data. The sentence beginning with “More precisely,” misses a verb and the semicolon doesn't really fit. I have the feeling that the sentences of the passage aren't really in the order they were meant to be…
Best,
the team Darmstadt
Marcus Waurick, 2020/01/30 15:52
Dear Gabriel, dear Darmstadt team,
thank you pointing out these mishaps.
Concerning your second point, we should have written something along the following lines:
More precisely, let M:dom(M)⊆C→L(L2(Ω)×L2(Ω)d) with sb(M)<ν0 for some ν0∈R. We assume that M satisfies the positive definiteness condition in Theorem 6.2.1, that is, there exists c>0 such that for all z∈CR≥ν0 we have RzM(z)≥c.
For ν≥ν0 we want to solve …
As you correctly pointed out, we really tried to convey the impression of treating other inhomogeneous b.v.p.s. Does putting `inhomogeneous' right before `other' resolve the issue?
Best regards,
MM
Jürgen Voigt, 2020/01/27 20:45
Dear virtual lecturers,
here are some comments:
three lines below the end of the proof of Theorem 12.13: I think one should say `… define the space of all …' (What is the set is clear: ran(γ).) But anyhow, I am very grateful for the definition! I had seen this space every now and then, but never seen the definition. But let me ask you: is this also some interpolation space; it is tempting to think this, because of the notation. (And if “yes”, then I suggest you should mention it.)
Concerning (12.1): At the first glance, it looks like a misprint. But here you see that it is rather unfortunate to use the same symbol for the sesquilinear form in Hilbert space and the bilinear form for dual pairs.
On the second line of the proof of Theorem 12.2.2 it seems minimally inconsistent not to write f∗(ˇx,0), but γf∗(ˇx). And then spelling out ed again seems unnecessary to me.
line -4 of the proof of Theorem 12.2.2: I think it is better to put a comma between `dense' and `as'; otherwise one first has to read on, in order to understand where `as' belongs. And in the following line one could spend `formula (12.1) …'
In Corollary 12.2.3 there should be a comma between `γf=0' and `and'; again in order to clarify what belongs where.
Seeing the first equation in Section 12.5, I was somewhat confused, because in the Robin boundary conditions for the heat equation there is no imaginary unit, and in fact everthing works also for the real case. Also, the properties in Sections 12.1 up to 12.4 should also work in the case of real scalars. (Maybe one could mention this somewhere.) But then I remembered that otherwise in this ISem it is quite essential to work with complex scalars, and also I now understand the more general notion of `Robin boundary condition'.
p.155, last two lines. One could write H−1/2 on line -2, and H1/2 on line -1. (Writing L2 twice seems confusing to me.)
Miscellanea:
p. 147, line -7: I rather doubt that a theorem can `readily imply' something.
p. 149, line -8: `Similarly' instead of `similar'?
p. 149, last sentence: This seems too “sketchy” to me. (Just try to start
reading with this sentence, which does not have a verb!)
Proposition 12.4.1, last line: comma between `unitary' and `and'.
The sentence after the proof of Proposition 12.4.2 is rather garbled: in particular the `being' is terribly dangling.
Altogether a very interesting and well-made lecture! Thanks!
Best wishes, Jürgen
Sascha Trostorff, 2020/01/28 10:43
Dear Jürgen,
thanks a lot for your valuable comments.
Concerning your question on the space H1/2: The quotient norm defined in the lecture seems not to be the standard norm on H1/2. In “Necas et al. Direct Methods in the Theory of Elliptic Equations.” one finds the definiton
∥u∥2H1/2(Ω):=∥u∥2L2(Ω)+∫∂Ω∫∂Ω|u(x)−u(y)|2|x−y|ddxdy,
which yields an equivalent norm. I don't know, whether this norm allows to identify H1/2 as a certain interpolation space.
Concerning formula (12.1): Yes, I agree and I am also not happy with this notation. We will think of an alternative for the final version.
Concerning your remark on the proof of Theorem 12.2.2, you are right, we just have forgot to rewrite this from an earlier version.
Concerning the Robin boundary condition: The imaginary unit is used to obtain a skew-selfadjoint operator A in Theorem 12.5.1. If one likes to deal with real coefficients instead, on has to formulate Picard's Theorem for accretive operators A, which also works (maybe this is a nice idea for a project, also to cover certain nonlinear problems).
Best regards
Sascha
Marcus Waurick, 2020/01/28 11:13, 2020/01/30 13:18
Dear Jürgen,
thank you very much for your comments. The fact that H1/2(∂Ω) can be viewed as an interpolation space (complex method) is provided in Chapter 7 and 8 of the book
Lions, J. L., & Magenes, E. (1972). Non-Homogeneous Boundary Value Problems and Applications. Volume 1.
You are right, it is best to mention this in the final version of the lecture notes. In the current version, we refrained from doing this because – as Sascha pointed out in his post – the norm we have used there does not seem to be the `standard' norm for H1/2, yet they are equivalent anyway.
Best regards,
Marcus
discussion/lecture_12.txt · Last modified: 2019/10/21 17:02 by matcs
Discussion on Lecture 12
Dear virtual lecturers,
comments from our session in Dresden:
p. 149, line 6: The second and third term should carry a minus sign.
It occured to me that for some participants it must be difficult to absorb the definition of H−1/2 (in Theorem 12.2.2). After all, Hilbert spaces are self-dual. Probably one should insert some explanation concerning Gelfand triples.
In the very last part of the proof of Corollary 12.2.3, we did not understand why the property div0q=divq is mentioned.
Best wishes, Jürgen
Dear Jürgen,
thank you for your comments!
p 149, l 6: You are right, thanks!
We should mentioned the notion of Gelfand triples at least in a remark somewhere. This is true. I think we should place this remark already in Lecture 9, where we introduced for instance H−1(Ω).
In the last part of Corollary 12.2.3 we sticked too much to the definition of the adjoint: After all we have an equation of the form ⟨p,f⟩+⟨q,gradf⟩=0for some p valid for all f∈dom(grad). Thus, q∈dom(grad∗) and grad∗f=p. We have applied this to p=divq. But you are right, this was nothing we wanted to show really, as the only thing needed was to assert q∈dom(grad∗). Thanks!
Best regards,
MM
Dear all,
as Jürgen already pointed out most, we've only got two grammatical remarks left:
Best,
the team Darmstadt
Dear Gabriel, dear Darmstadt team,
thank you pointing out these mishaps.
Concerning your second point, we should have written something along the following lines:
More precisely, let M:dom(M)⊆C→L(L2(Ω)×L2(Ω)d) with sb(M)<ν0 for some ν0∈R. We assume that M satisfies the positive definiteness condition in Theorem 6.2.1, that is, there exists c>0 such that for all z∈CR≥ν0 we have RzM(z)≥c. For ν≥ν0 we want to solve …
As you correctly pointed out, we really tried to convey the impression of treating other inhomogeneous b.v.p.s. Does putting `inhomogeneous' right before `other' resolve the issue?
Best regards,
MM
Dear virtual lecturers,
here are some comments:
three lines below the end of the proof of Theorem 12.13: I think one should say `… define the space of all …' (What is the set is clear: ran(γ).) But anyhow, I am very grateful for the definition! I had seen this space every now and then, but never seen the definition. But let me ask you: is this also some interpolation space; it is tempting to think this, because of the notation. (And if “yes”, then I suggest you should mention it.)
Concerning (12.1): At the first glance, it looks like a misprint. But here you see that it is rather unfortunate to use the same symbol for the sesquilinear form in Hilbert space and the bilinear form for dual pairs.
On the second line of the proof of Theorem 12.2.2 it seems minimally inconsistent not to write f∗(ˇx,0), but γf∗(ˇx). And then spelling out ed again seems unnecessary to me.
line -4 of the proof of Theorem 12.2.2: I think it is better to put a comma between `dense' and `as'; otherwise one first has to read on, in order to understand where `as' belongs. And in the following line one could spend `formula (12.1) …'
In Corollary 12.2.3 there should be a comma between `γf=0' and `and'; again in order to clarify what belongs where.
Seeing the first equation in Section 12.5, I was somewhat confused, because in the Robin boundary conditions for the heat equation there is no imaginary unit, and in fact everthing works also for the real case. Also, the properties in Sections 12.1 up to 12.4 should also work in the case of real scalars. (Maybe one could mention this somewhere.) But then I remembered that otherwise in this ISem it is quite essential to work with complex scalars, and also I now understand the more general notion of `Robin boundary condition'.
p.155, last two lines. One could write H−1/2 on line -2, and H1/2 on line -1. (Writing L2 twice seems confusing to me.)
Miscellanea:
p. 147, line -7: I rather doubt that a theorem can `readily imply' something.
p. 149, line -8: `Similarly' instead of `similar'?
p. 149, last sentence: This seems too “sketchy” to me. (Just try to start reading with this sentence, which does not have a verb!)
Proposition 12.4.1, last line: comma between `unitary' and `and'.
The sentence after the proof of Proposition 12.4.2 is rather garbled: in particular the `being' is terribly dangling.
Altogether a very interesting and well-made lecture! Thanks!
Best wishes, Jürgen
Dear Jürgen,
thanks a lot for your valuable comments.
Concerning your question on the space H1/2: The quotient norm defined in the lecture seems not to be the standard norm on H1/2. In “Necas et al. Direct Methods in the Theory of Elliptic Equations.” one finds the definiton ∥u∥2H1/2(Ω):=∥u∥2L2(Ω)+∫∂Ω∫∂Ω|u(x)−u(y)|2|x−y|ddxdy, which yields an equivalent norm. I don't know, whether this norm allows to identify H1/2 as a certain interpolation space.
Concerning formula (12.1): Yes, I agree and I am also not happy with this notation. We will think of an alternative for the final version.
Concerning your remark on the proof of Theorem 12.2.2, you are right, we just have forgot to rewrite this from an earlier version.
Concerning the Robin boundary condition: The imaginary unit is used to obtain a skew-selfadjoint operator A in Theorem 12.5.1. If one likes to deal with real coefficients instead, on has to formulate Picard's Theorem for accretive operators A, which also works (maybe this is a nice idea for a project, also to cover certain nonlinear problems).
Best regards
Sascha
Dear Jürgen,
thank you very much for your comments. The fact that H1/2(∂Ω) can be viewed as an interpolation space (complex method) is provided in Chapter 7 and 8 of the book
Lions, J. L., & Magenes, E. (1972). Non-Homogeneous Boundary Value Problems and Applications. Volume 1.
You are right, it is best to mention this in the final version of the lecture notes. In the current version, we refrained from doing this because – as Sascha pointed out in his post – the norm we have used there does not seem to be the `standard' norm for H1/2, yet they are equivalent anyway.
Best regards,
Marcus