Dear all,

regarding the last step in the proof of Lemma 5.8, namely the convergence $u_n\to u$, isn't this just the usual argument that $B_n{f}_n\to Bf$ for a sequence $\left(B_n\right)$ of operators between to Banach spaces that converges strongly to $B$ and a sequence $f_n$ that converges to $f$ in the norm of the first Banach space?

Best wishes, Hendrik

Hello,

In Lemma 5.9 we talk about a $w_0$ but it is nowhere defined, should that be a $w$?

Greetings Ragon

Dear authors,

in the proof of the Theorem 5.15 $\left(i^{\prime }\right)\Rightarrow \left(ii\right)$ shouldnt the contraction $C[0,\frac{1}{\alpha }]$ used instead of $C[0,\alpha ]$?

Best, Anna

Dear all,

I have a small question concerning Theorem 5.15. In condition (ii), is “for every” equivalent to “for some” $\alpha >0$?

Regards, Sahiba

Dear ISEM-Team,

I have a question concerning Exercise 7.1: Shouldn't we assume that $c=0$? Otherwise the statement seems to be wrong (just consider $X=\left\{x\right\}$ to be a singleton and $c\left(x\right)=1$).

Best regards

Sascha

Dear virtual lecturers,

in the discussions in the Darmstadt team we found several times that what we are truly missing is some sort of reference example of a graph, that helps to see the effects described in the text. In particular the behavior of solutions to the heat equation should be something that has an obvious physical intuition and it would be nice to see this in some concrete example.

I try to explain what I mean: I always try to compare the results with the situation in spatial domains that I know better and there for instance the domain monotonicity described in this lecture is something completely intuitive and it becomes also clear there why it only works for Dirichlet boundary conditions and not for Neumann conditions. I am still lacking an intuition what the “boundary conditions” mean here, as there is no apparent boundary in the graphs and I would hope to see the difference of these boundary conditions in a concrete example. Is something like this possible and will there be some examples in future lectures?

Finally one remark about the text of lecture 7: In the proof of Lemma 5.3 (b) (line 22 ff of page 85) you want to show that $x\mapsto e^{-tx}$ belongs to $A$ for all $t\ge 0$. As you later prove that $A$ equals $C_0\left(\right[0,\infty \left)\right)$ that cannot be true for $t=0$, as then this function is constantly one and thus does not belong to this space. In fact you probably mean that the assertion of the Lemma is obvious for $t=0$ and the argument given gives the assertion for all $t>0$.

Best regards,

Robert

Dear virtual lecturers,

it seems to me that some references on lecture 7 are misdirecting. In particular:

• P.84 at the end of the second paragraph of the proof, Proposition 5.2 b) is being cited, which does not exist. We only have Lemma 5.2 without enumerated items.
• P.84 at the beginning of the third paragraph, Proposition 5.2 c) is being cited, which does not exist either.
• P.86 after the proof of (b), we cite Lemma 5.2 c), while Lemma 5.2 has no enumerated items.
• P.86 right below remark 5.5, Lemma 5.2 b) is being cited.
• P.87 on the first line, Lemma 5.2 b) again.

I'm guessing, that Lemma 5.2 on domain monotonicity covers the statement of lost Lemma 5.2 c), while b) covers the Markov property. What could have been Lemma 5.2 a) then?

Best regards, Johannes

Dear Paco,

You can of course send your solutions to us by e-mail. We will check them briefly and then publish them on the web page.

Best, Christian

Dear Authors,

In team Hagen, we are solving at least two of the exercises proposed for each lecture each week.

Would be OK if we post our solutions to the exercises for previous lectures? This is a unique opportunity to receive feedback, see different solution methods, and we could also help the teams responsible for the exercises . (Currently, only the exercises for Lecture 1 have been posted. )

What do you think?

Best Regards,

Paco