Timeline for How to use blow-up to prove the boundary regularity for a harmonic function
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 2, 2021 at 14:23 | comment | added | Holden Lyu | @RegaurCarchern You are welcome. | |
| Oct 2, 2021 at 14:22 | comment | added | Luis Yanka Annalisc | @user734979, Thank you very much. | |
| Oct 2, 2021 at 14:19 | comment | added | Holden Lyu | @RegaurCarchern For the first one, I think you may be confused about the covering gap. At the boundary we may not get the covering by small balls. But, as I remember, you can get a similar covering lemma leave the interior part for the interior estimate. For the second one, I think you can use a sequence smooth functions tending to the given condition in the C^{k} norm and you got the estimate for the solution. With the uniqueness of solution you can show that the weak solution is C^{k+2} | |
| Oct 2, 2021 at 13:40 | comment | added | Luis Yanka Annalisc | @user734979, I mean that I cannot get the priori estimates by using the blow up method and also have trouble in showing that the weak solution is in $ C^{k+2} $. | |
| Oct 2, 2021 at 13:26 | comment | added | Holden Lyu | @RegaurCarchern Well, I’m kind of confused about what you are saying. Do you mean you’ve already get the apriori estimate and you don’t know how to derive the regularity on the boundary? | |
| Oct 2, 2021 at 12:20 | comment | added | Luis Yanka Annalisc | @user734979 I am reading the same book too, and I still feel confused about the statement on the book. How can I use the priori estimates on it and how can I deal with the boundary value? | |
| Sep 1, 2021 at 12:50 | comment | added | Holden Lyu | @GiorgioMetafune Thank you very much for your help. I’m gonna try to do it in this method and I believe this is the right way. | |
| Aug 31, 2021 at 19:15 | comment | added | Giorgio Metafune | In my experience, it is easier to prove Schauder estimates for the Laplacian (or $D_t-\Delta$) in the whole space by blow-up and then use standard methods for variable coefficients and local estimates. If you look at the web, there is a short class note by R. Halshofer yielding Schauder estimates for $\Delta$ and the method generalizes to $D_t-\Delta$. Concerning boundary estimates, it is easier to go to the half space by local coodinates, to reduce to a homogenuous boundary value problem (by subtracting an extension of the boundary value) and to reflect to go to the whole space. | |
| Aug 30, 2021 at 16:40 | comment | added | Holden Lyu | @GiorgioMetafune The mollification may not be used near the boundary. But I’ve just reviewed the proof of cor. 2.16 and I think maybe we can use the mollification inside to get that the solution is $C^{k, \alpha}$ in any subset and therefore in $\Omega$. So we can use the a-priori estimate. | |
| Aug 29, 2021 at 8:38 | comment | added | Giorgio Metafune | Sorry, I just noticed that the book I recommended is exactly that you are reading (you did not write the authors and I got confused). I agree, the argument needs smooth functions and, in principle, is an a-priori estimates. However, having it, it can be used on a single equation, via approximation. See the proof of corollary 2.16 (in the edition I have), where the estimates are applied to mollifications of the weak solution.. | |
| Aug 28, 2021 at 19:38 | comment | added | Holden Lyu | @GiorgioMetafune But here we only assume that $u\in H^1$ so we may need to do it by Green function and all the techniques we talked above should be used for the general equation by the method of continuity | |
| Aug 28, 2021 at 19:36 | comment | added | Holden Lyu | @GiorgioMetafune I think you are right. But when I’m trying to write it down I’m faced with another problem: all the blow-up trick are based on one assumption—the solution is $C^{k, \alpha}$, which allows to consider $u-p$ where $p$ is a polynomial determined by the k-th derivative. And we would use the continuity method to determine the uniqueness of solution and therefore all the estimate can be done. | |
| Aug 28, 2021 at 14:28 | comment | added | Giorgio Metafune | Assme $k=2$. You can change coordinates from $\Omega$ to the half plane using the smoothness of the boundary. Then you get another ellipitc equation in the half-plane with Holder coefficients. This is the way how the smoothness of the boundary is generally used. I do not know if the authors had that in mind. | |
| Aug 27, 2021 at 23:01 | comment | added | Holden Lyu | @GiorgioMetafune Thank you very much for this reference but I’m still confused with a tricky problem: on the boundary condition we also assumed the boundary to be $C^{k, \alpha}$ and where would this assumption be used? I’m thinking that there may be some transform wish can remain the bound of the norm of C^{k, \alpha}$ but I can’t write it down clearly. | |
| Aug 26, 2021 at 8:11 | comment | added | Giorgio Metafune | You can have a look at Regularity Theory for Elliptic PDE by Xavier Fernandez-Real and Xavier Ros-Oton (you can download it). In Chapter 2, the second proof of Proposition 2.25 they explain well the blow-up method in the case of interior points | |
| Aug 26, 2021 at 3:35 | comment | added | Math604 | I recall seeing a paper (and he also had some notes) of Leon Simon from 90's that had a proof using blow up analysis. | |
| Aug 25, 2021 at 22:02 | review | First posts | |||
| Aug 26, 2021 at 3:10 | |||||
| Aug 25, 2021 at 21:56 | history | asked | Holden Lyu | CC BY-SA 4.0 |