Timeline for Carleman estimates on monotonicity formulas
Current License: CC BY-SA 3.0
17 events
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| Jun 11, 2014 at 8:25 | comment | added | Willie Wong | @ChristianClason: those are some nice references. Thanks! | |
| Jun 6, 2014 at 11:33 | comment | added | user51604 | Great answers guess is a good idea to follow the literature provided :), yea i was wondering if the Carleman estimates effectively bound the inner norm with some boundary estimates. The jump situation would be really helpful, though i thing the really main problem is somehow how "diagonal" are the operators, which as far as i seen not only matters for the eigenvalue technique but also for the carleman estimate approach (ams.org/journals/tran/2010-362-01/S0002-9947-09-04799-0 Koch and Colombini) | |
| Jun 6, 2014 at 9:31 | comment | added | Christian Clason | Regarding Carleman estimates for coupled systems: This seems to be the main current direction for people working on Carleman estimates, especially in the inverse problem and controllability communities; you might want to look there for ideas (see, e.g., Masahiro Yamamoto's survey paper (Section 7.8) or this paper by Jean-Pierre Raymond). | |
| Jun 6, 2014 at 9:20 | comment | added | Christian Clason | ...and especially at Nicolas Lerner's lecture notes mentioned in a comment there; Chapter 2 deals with elliptic operators with jumps in the principal coefficient. | |
| Jun 6, 2014 at 7:49 | comment | added | Willie Wong | You may want to look at this thread, which may tell you whether you will find the estimates useful. Two other side notes: most versions of Carleman estimates I am aware of requires the operator to have smooth coefficients, and off the top of my head I am not 100% sure whether you have a chance with discontinuous coefficients to even use those inequalities. Also, it is notoriously difficult to get Carleman estimates for coupled systems: most proofs I have seen only apply to (effectively) scalar equations. | |
| Jun 6, 2014 at 7:43 | comment | added | Willie Wong |
Okay, I see better now what you are doing. The equation that I just tagged as (**) is in the general shape of a Carleman estimate: it controls the $L^p$ norm of some interior quantity with the $L^p$ norm of some boundary quantity. The main difference is that usually Carleman estimates comes with weights (sometimes blowing up at the boundary). So you will have to modify the argument a little bit to control the weights.
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| Jun 6, 2014 at 7:39 | history | edited | Willie Wong | CC BY-SA 3.0 |
tagged a couple equations so we can refer to them.
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| Jun 5, 2014 at 16:10 | comment | added | user51604 | I updated the thread :) | |
| Jun 5, 2014 at 16:09 | history | edited | user51604 | CC BY-SA 3.0 |
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| Jun 5, 2014 at 10:29 | history | edited | user51604 | CC BY-SA 3.0 |
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| Jun 5, 2014 at 10:25 | comment | added | user51604 | Totally right, sorry for the typos and omissions (first post). It should make sense now, and yes I forgot to mention we deal with critical points, i.e. solutions to a particular divergence form elliptic equation. About the technique I will post it later, as it requires a little bit of space and im short of time :) | |
| Jun 5, 2014 at 10:22 | history | edited | user51604 | CC BY-SA 3.0 |
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| Jun 4, 2014 at 11:52 | comment | added | Willie Wong | Also, do you mean $e(\nabla u)$ instead of $\nabla e(u)$? The latter doesn't make sense as $u$ is a vector and $e$ acts on square matrices. Are you also assuming that your $u$ solves some sort of an equation? Or perhaps $A$ has certain ellipticity? For arbitrary $u$ I cannot see any estimates being possible for $f(r)$, even in the scalar case. And in the elliptic case then the monotonicity is trivially true for $\alpha \leq 0$, so perhaps you have a certain rate in mind for your $\alpha$? | |
| Jun 4, 2014 at 11:49 | comment | added | Willie Wong | Can you indicate some sources for the "several techniques" that one can apply in the scalar case? (I just want to see if an example will jog my memory a bit.) | |
| Jun 4, 2014 at 11:49 | history | edited | Willie Wong |
edited tags
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| Jun 4, 2014 at 11:12 | review | First posts | |||
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| Jun 4, 2014 at 10:56 | history | asked | user51604 | CC BY-SA 3.0 |