Timeline for Epsilon regularity: what does it say and where does it come from?
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
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| Apr 15, 2017 at 17:38 | comment | added | Deane Yang | See the work of Aaron Naber with collaborators on recent work in this direction. For example, arxiv.org/abs/1403.4176 | |
| Sep 21, 2010 at 16:35 | answer | added | Terry Tao | timeline score: 16 | |
| Sep 20, 2010 at 19:10 | history | edited | hce | CC BY-SA 2.5 |
Added a fourth question and the tag geometric-analysis.
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| Sep 19, 2010 at 17:01 | comment | added | hce | About conformal invariance: I call the quantity $r^{2-k}\int_{B_r}|\nabla u|^2$ the conformally invariant energy. (I now notice that I called this a rescaling of the energy, which does not sound correct; it is definitely a scaling invariant quantity though) | |
| Sep 19, 2010 at 15:38 | comment | added | Rbega | Yes, the pointwise estimate is the estimate of the "gradient" at zero (the point is the center of the ball (and hence the center $0$) is arbitrary. One other thing, the conformal invariance is a special feature of harmonic maps from surfaces (i.e. 2-dimensional domains). There is a scaling invariance for higher dimensions but this is different. | |
| Sep 19, 2010 at 8:40 | comment | added | hce | This is why I find it interesting to understand what exactly $\varepsilon'$ is. | |
| Sep 19, 2010 at 8:39 | comment | added | hce | When you said '$\varepsilon$-regularity quantifies how the sequence can fail to smoothly converge' you reminded me of Tristan Rivière ( arxiv.org/abs/math/0304396 ), who calls the Schoen-Uhlenbeck lemma 'the earliest example of energy quantization in non-linear analysis'. I guess this means that once we know that the conformally invariant energy is less than $\Lambda$, but $\Lambda$ is greater than $\varepsilon'$ (the sup of all $\varepsilon$ for which regularity is guaranteed), then we can expect to have a singular set (I admit this is quite tautological). | |
| Sep 19, 2010 at 8:37 | comment | added | hce | Thanks! Regarding your first comment: is the pointwise estimate you refer to the estimate on the derivative at zero? | |
| Sep 19, 2010 at 3:47 | answer | added | Deane Yang | timeline score: 16 | |
| Sep 19, 2010 at 1:45 | comment | added | Rbega | I should add another advantage of considering classical solutions is that there are then really slick proofs of $\epsilon$-regularity type theorems. A good example is the proof of the Choi-Schoen theorem or of the smooth version of Allard regularity both of which are (I believe) in Colding and Minicozzi's book in not in the "excursion". These sorts of proofs might may also provide you with some intuition as they strip out all the technicalities and really get at the essence. | |
| Sep 19, 2010 at 1:42 | answer | added | Evan Wright | timeline score: 3 | |
| Sep 19, 2010 at 1:40 | comment | added | Rbega | This isn't an answer, but one thing that may be helpful is to not consider the $\epsilon$-regularity lemma for classical solutions $u_i$. In this case the smoothness is not an issue (as you get it "standard elliptic estimates" i.e. Schauder theory) but rather the point is the uniformity (independent of the $u_i$) of the pointwise estimate. In particular, let $u_i$ have uniformly bounded energy. A subsequence will converge weakly to a weak solution $u$. $\epsilon$-regularity quantifies how the sequence can fail to smoothly converge. Namely, energy concentrating on small scales. | |
| Sep 18, 2010 at 16:56 | history | asked | hce | CC BY-SA 2.5 |