Timeline for Prove of the shape-derivative identity relating the shape and material derivative of a shape-dependent function
Current License: CC BY-SA 4.0
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| Sep 7, 2020 at 10:17 | comment | added | 0xbadf00d | It took me a while before I could come back to this question. I think it's worth considering the pointwise analogue of the material/shape derivative which is considered here. I've asked a separate question for that: mathoverflow.net/q/371076/91890. It would be great if you could take a look. | |
| Sep 7, 2020 at 10:16 | vote | accept | 0xbadf00d | ||
| Aug 3, 2020 at 18:06 | history | undeleted | DCM | ||
| Aug 1, 2020 at 10:02 | history | deleted | DCM | via Vote | |
| Jul 27, 2020 at 19:11 | history | edited | DCM | CC BY-SA 4.0 |
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| Jul 26, 2020 at 12:02 | comment | added | DCM | Taking $C^\infty_c(D)$ as the space of test functions seems sensible because it gives you a way to take linear combinations and limits of the $y(\Omega_t)$ in $\mathscr{D}'(D)$. The other way natural way to do this is to choose your $E$ functor $\Omega\mapsto E_\Omega$ to be one for which $E_\Omega = \{f_{|\Omega}: f\in E_D\}$ for all $\Omega\in \mathcal{A}$, at which point you can do the same in $E_D$ (although this latter approach requires you to check that your limits are independent of which extensions you choose - your reference does it like this). | |
| Jul 25, 2020 at 18:36 | comment | added | 0xbadf00d | Side note: You've used $\varphi\in C_c^\infty(D)$ instead of $\varphi\in C_c^\infty(\Omega)$. Could you elaborate on why this is important or at least useful? | |
| Jul 25, 2020 at 18:33 | comment | added | 0xbadf00d | Now you've chosen $E_\Omega=L^1_{\text{loc}}(\Omega)$ and I guess that weak convergence of a sequence $(f_n)_{n\in\mathbb N}\subseteq L^1_{\text{loc}}(\Omega)$ to $f\in L^1_{\text{loc}}(\Omega)$ holds if and only if $$\int f_n\varphi\:{\rm d}\lambda^{\otimes d}\xrightarrow{n\to\infty}\int f\varphi\:{\rm d}\lambda^{\otimes d}\tag{12}$$ for all $\varphi\in C_c^\infty(\Omega)$ (I don't remember whether we need additional assumptions to show that). So, while useful for its own, the question for the "strong" derivatives in my post is still open. | |
| Jul 25, 2020 at 18:33 | comment | added | 0xbadf00d | Thank you for your answer. I think what you're doing is considering the "weak" versions of the material and shape derivative. They are defined in the same way as in the question, but the limits in the definition of the Fréchet derivatives have to be understood with respect to the weak topology. So, for example, $$\left\langle\frac{y(\Omega_t)\circ\left.T_t\right|_{\Omega}-y(\Omega)}t,\varphi\right\rangle\xrightarrow{t\to0+}\left\langle\dot y(\Omega;v),\varphi\right\rangle\tag{11}$$ for all $\varphi\in E_\Omega'$. | |
| Jul 25, 2020 at 18:07 | history | edited | DCM | CC BY-SA 4.0 |
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| Jul 25, 2020 at 15:02 | history | answered | DCM | CC BY-SA 4.0 |