Timeline for Euler-Lagrange Equation and "Eigen Value "
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| S Mar 16, 2015 at 16:02 | history | suggested | Hachino | CC BY-SA 3.0 |
Improved formatting overall, corrected typos here and there.
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| Mar 16, 2015 at 15:57 | review | Suggested edits | |||
| S Mar 16, 2015 at 16:02 | |||||
| Mar 16, 2015 at 15:46 | review | Close votes | |||
| Mar 16, 2015 at 17:01 | |||||
| Mar 15, 2015 at 11:06 | answer | added | Hachino | timeline score: 1 | |
| Mar 15, 2015 at 10:17 | comment | added | Hachino | Give me a few minutes, I was going to do so anyway. | |
| Mar 15, 2015 at 10:16 | comment | added | Learner | @Hachino I would be glad if you could check my calculations and some further guidance . | |
| Mar 15, 2015 at 10:00 | comment | added | Learner | @Hachino Yes, my main objective is to understand the variations of EV/EF's wrt the domain and shape optimisation . | |
| Mar 15, 2015 at 9:53 | comment | added | Hachino | Btw, your notations and computations suggest that you are trying to understand variations of eigenvalues/eigenfunctions w.r.t. the underlying domain and shape optimization. Is it so ? | |
| Mar 15, 2015 at 9:51 | comment | added | Hachino | Your definition of $\lambda(\Omega_t) := \mathbf{min} \{R(v, \Omega_t) : v \in H^{1,2}(\Omega_t)\}$ implies that 1) $\lambda$ is nonnegative 2) any constant, nonzero function achieves a zero value for the Rayleigh quotient, which means that $\lambda = 0$. But under an additional condition, for instance $\lambda(\Omega_t) := \mathbf{min} \{R(v, \Omega_t) : v \in H^{1,2}(\Omega_t), \int_{\Omega_t} v = 0\}$ (or more generally, that your $v$'s are orthogonal to some subset of eigenfunctions of the Neumann laplacian on $\Omega_t$), $\lambda$ acquires a nontrivial value. | |
| Mar 15, 2015 at 9:46 | comment | added | Learner | @Hachino Sir, The curious factor $\mu(u)$ does appear because of the differentiation of $\left(\int_{\Omega_t }(u+s\varphi)^q\right)^{2/q}$ w.r.t $s$ . I do not understand what orthogonality condition you mean . I would be glad if you could elaborate it for me . | |
| Mar 15, 2015 at 9:33 | comment | added | Hachino | I would have two things to point out : first, if you set $\alpha = 0$ and $q=2$, your final equations are the same as in the first case, apart from this curious $\mu(u)$, which probably shouldn't be there. Secondly, you forgot to add orthogonality conditions in both cases, in the spirit of $\int_{\Omega_t} u = 0$. Otherwise, non zero constant functions are obvious global minimizers. | |
| Mar 15, 2015 at 9:00 | review | First posts | |||
| Mar 15, 2015 at 9:17 | |||||
| Mar 15, 2015 at 8:58 | history | asked | Learner | CC BY-SA 3.0 |