Timeline for A basic stability question
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Aug 26, 2018 at 1:23 | comment | added | Willie Wong | @MathStudent: Not sure. If your assumed bound on $V_2 - V_1$ were in $L^\infty$ and not $L^2$, then ODE arguments will tell you that the level sets of $u_1$ and $u_2$ are similar, which coupled with $L^\infty$ bounds on $\nabla u_i$ would (probably) imply using a simple triangle inequality argument that $\nabla u_1 - \nabla u_2$ is controlled (even in $L^\infty$). It is not immediately clear to me whether in $L^2$ (and in higher dimensions) there may be more delicate counterexamples. | |
| Aug 25, 2018 at 16:24 | comment | added | A random mathematician | @ Willie Wong: Do you think the result may hold if we additionally assume that $|\nabla u_i|$ are bounded( i.e. $\lambda \leq |\nabla u_i| \leq \lambda$)? | |
| Aug 16, 2018 at 22:54 | history | edited | Willie Wong | CC BY-SA 4.0 |
added 957 characters in body
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| Aug 16, 2018 at 21:48 | comment | added | A random mathematician | You are absolutely right. Thank you, Willie. | |
| Aug 16, 2018 at 21:48 | vote | accept | A random mathematician | ||
| Aug 16, 2018 at 21:28 | comment | added | Willie Wong | @MathStudent: your computation for $\nabla u_k$ cannot be right. By definition you have that $\nabla u_k(x,y) = k^3 / (1 + k^6 x^2) \partial_x$. Evaluating it at $(-1/2k,0)$ you should get something of the order $k^3 / (1 + k^4)$ which is approximately $1/k$. The value of $\nabla v_k(x,y)$ is hard to compute exactly, but the key idea is that at $(-1/2k,0)$, $v_k = 0$, and hence its derivative is expected to be of order $k^3$. You should double check your computations again. | |
| Aug 16, 2018 at 20:29 | comment | added | A random mathematician | I doubled checked the computations. It seems to me that $\nabla v_k \approx \frac{1}{k^3}$ and $\nabla u_k \approx \frac{1}{k^3}$. So this doesn't seem to provide a counterexample. | |
| Aug 16, 2018 at 4:20 | vote | accept | A random mathematician | ||
| Aug 16, 2018 at 20:27 | |||||
| Aug 15, 2018 at 18:14 | history | answered | Willie Wong | CC BY-SA 4.0 |