Discussion Board Lecture 07
Misprints in the proof of Theorem 7.15
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The form $ a $ in the proof should be given by $ a(u,v):=\int_{\Omega}{\nabla u \cdot \overline{\nabla v}}+\int_{\partial\Omega}{\beta u\overline{v}}. $ The conjugates are missing. Furthermore, there is a superfluous "Let $ 0<\varepsilon<1 $." in the proof.
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Dear Karsten thanks a lot for pointing this out; the form should take its values in $ \mathbb K $. The first mistake is a remnant from a previous version where the problem was only treated for the real case, the $ \epsilon $ is a relict from a previous version. Best wishes, Jürgen
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Dear ISem team, one line after the epsilon: Euclid's inquality of course also holds for $ a=0 $ or $ b=0 $, which is maybe needed one line later on. Best wishes
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Dear Johannes, you're right, of course – thanks a lot for pointing this out! Best wishes, Hendrik
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Does symmetry of forms imply self-adjointness of operators?
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Dear ISEM-Team, in the Proof of Theorem 7.15 you write that that the Robin Laplacian is self-adjoint because the associated form is symmetric. My question is, is that really enough to show the self-adjointness? In theorem 6.10 we required that the form is symmetric AND coercive to conclude the self-adjointness. I'm pretty sure that if the operator in questions is H-elliptic like in our case, you can do basically the same argument as in 6.10 to show the self-adjointness but is it also enough if the operator is just symmetric? Best Regards, Clemens Bombach
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Dear Clemens, you are definitely correct that symmetry of an operator does not imply self-adjointness. (This was not our argument!) Your reference to Theorem 6.10 is quite correct. It implies that $ A+\omega I $ is self-adjoint, and then $ A $ is self-adjoint. Best wishes, Jürgen
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Two remarks
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Dear all, I think in Theorem 7.4 (Gauss) one should assume that $ \Omega $ is bounded or that the function $ u $ has compact support. Furthermore one should probably briefly explain why the weak normal derivative is unique, i.e. why is the set of boundary values a sufficiently large subset of $ L_2(\partial \Omega) $. Best wishes André
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Dear André, at the very beginning of section 7.1 it is assumed that $ \Omega $ is bounded throughout the entire section. Nevertheless, I think it will be quite convenient for the reader if this assumption is repeated within the formulation of the theorem as well as the precise assumptions concerning the boundary regularity (of course, the discussion preceding Gauss' Theorem indicates that $ \Omega $ is supposed to have $ C^1 $-boundary, but it is not explicitly stated). Concerning your second suggestion, I don't have a hunch whether or not it is that easy to explain briefly, but also fairly comprehensibly the uniqueness of the weak normal derivative without assuming some familiarity with results from topology and topological measure theory. For instance, this is my explanation (but maybe there is a much simpler way): Let $ \Omega $ be bounded with $ C^1 $-boundary. Consider $ A:=\operatorname{tr}(C^1(\overline\Omega)) $. Then $ A $ is a unital subalgebra of $ C(\partial\Omega) $ separating the points of $ \partial\Omega $ and (in the case $ \mathbb K=\mathbb C $) closed with respect to complex conjugation. Thanks to the Stone-Weierstraà Theorem, $ A $ is dense in $ C(\partial\Omega) $ (with respect to the uniform norm) and the latter space is dense in $ L_2(\partial\Omega,\mu) $ for each finite Borel measure $ \mu $ on $ \partial\Omega $. Having this at disposal, it is easy to conclude the uniqueness: Let $ h,\widetilde h $ be two weak normal derivatives (say for $ u\in H^1(\Omega) $) in $ L_2(\partial\Omega) $. Then in particular $ h-\widetilde h\in A^\perp $, hence $ h-\widetilde h\in C(\partial\Omega)^\perp=\{0\} $, which yields $ h=\widetilde h $ (in $ L_2(\partial\Omega) $) as claimed. Best wishes, Heiko P.S.: By the way, there is a misprint in the sentence right before Theorem 7.9: One should read $ L_2(\partial\Omega) $ instead of $ L_2(\Omega) $.
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Dear André, and dear Heiko, thanks for pointing out the inconveniences connected with not always stating the precise assumptions, and thanks for pointing out and mending the gap. We simply overlooked to care about the uniqueness of the normal derivative; sorry! And in Section 8.1 you may have seen that we use precisely the argument of Heiko for the denseness of the range of the trace. And thanks for the misprint. Best wishes, Jürgen
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Comments on the proofs of Thm. 7.10 and 7.11
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Dear ISem team, during our discussion in Karlsruhe it was remarked that in the third to the last line in the proof of Thm. 7.10 you use the first of Green's formulas for a function in $ H^1 $, but in (7.1) it was only for $ C^1 $-function instead of it. (Of course, the weaker formulation can be shown by density.) In the proof of Thm. 7.11 it remained unclear to us how you integrate over $ W_k $. This seems to have been done after the longest displayed line. The problem is that$ W_k $ is no subset of a hyperplane in $ \mathbb{R}^n $, but a "curved" set. Best wishes
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Dear Johannes, now you know why we wrote `we will be somewhat sketchy in the presentation' in the introduction to the interlude :-) As for the third to the last line in the proof of Thm. 7.10: yes, we use a density argument there. That's why we referred to Theorem 7.9, which provides the required continuity to make the density argument work. The proof of Thm. 7.11 is a bit more sketchy at the point where we integrate over $ U_{k,\varepsilon} $. At no point do we integrate over $ W_k $; this is impossible since $ W_k $ is not just "curved", it need not even be a nice (Lipschitz) manifold! You should rather think of Cavalieri's principle. To be more precise: for simplicity let's suppose that $ y^k=e_n $, the $ n $-th unit vector. (You can produce this by a rotation and scaling.) Then    $ U_{k,\varepsilon} = \bigl\{(y,g_k(y)+s);\,y\in W_k',\ 0<s<\varepsilon\bigr\} $, with $ W_k'\subseteq\mathbb R^{n-1} $ and $ g_k\colon W_k'\to\mathbb R $ as in the description of a normal continous graph. In this parametrisation of $ U_{k,\varepsilon} $ you integrate first over $ s $ and then over $ y $ to obtain the desired estimate. Does this help? Best wishes, Hendrik
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