Timeline for Second order differentiability of solution operator to nonlinear boundary value problem
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Jan 17 at 11:20 | comment | added | Hannes | I see, the authors claim F-differentiability of $S \colon L^\infty(\Omega_T) \to W(0,T) \cap L^\infty(\Omega_T)$. That seems indeed not clear to me at all, anything involving $\partial_t p$ can only be modeled in $L^2(0,T;H^{-1})$ and I do not see how you would get $L^\infty(\Omega_T)$ from that. I suppose that is why the authors do the directional derivatives. One would need to bootstrap regularity and move to a more regular functional analytic setting if you ask me. | |
| Jan 17 at 10:42 | comment | added | Paul Joh | Good idea ! But I think for boundedness of the inverse, I need $\lVert \partial_t p - \Delta p + (2u-a+q)p \rVert_{\infty} \geq C \cdot \lVert p \rVert_{\infty}$ so an opposite bound | |
| Jan 17 at 9:12 | comment | added | Hannes | They have that covered in the paper, no? Lemma 3.2 and Theorem 3.1 with the appropriate substitutions? | |
| Jan 17 at 8:50 | comment | added | Paul Joh | Hey thanks, I looked at the implicit fct. theorem in your reference. Super interesting. But I think, the bottleneck is the requirement that $e_y$ has a bounded inverse. In my case, this is not clear for the map $p \mapsto \partial_{T} p - \Delta p + (2u-a+q)p$ in $L^{\infty}$ | |
| Jan 16 at 20:28 | comment | added | Hannes | You're welcome! Regarding differentiability, if in the above notation $e$ is $m$ times continuously Fréchet differentiable, then so is $S$. This is the magic of the implicit function theorem! (Here in fact $e$ is quadratic, so its third derivative vanishes already.) | |
| Jan 16 at 19:12 | comment | added | Paul Joh | Hey Hannes, thanks alot for your help ! I gained alot of intuition from your answer :) I also wondered how to prove the existence of the second order derivative rigorously. In your book it is just assumed. Unfortunately, here the implicit function theorem fails me (at least in the way its applied in Fredi Tröltzsch). | |
| Jan 16 at 16:12 | history | answered | Hannes | CC BY-SA 4.0 |