Timeline for Reference request: Is if possible to estimate the local behaviour of the solution of $\nabla \cdot a(x) \nabla f=g$ via constant coefficients?
Current License: CC BY-SA 4.0
9 events
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| May 11, 2021 at 15:14 | comment | added | Kernel | Thank you very much everyone for the references and comments, I think I misremembered something I saw in a seminar a few years ago and tried to apply in a rushed way. Your input is very appreciated. | |
| May 11, 2021 at 14:14 | comment | added | mlk | @Kernel I haven't read the book myself, but I think "Elliptic regulary theory" by Lisa Beck might be worth a read. At least from a quick glance there is a section on blow up and the book uses Morrey and Campanato spaces, which are extremely related to your definition of $M$. | |
| May 11, 2021 at 13:37 | comment | added | Kernel | @mlk Do you have a reference for this? | |
| May 10, 2021 at 19:43 | vote | accept | Kernel | ||
| May 10, 2021 at 19:07 | comment | added | mlk | In regularity theory of PDEs and for minimization problems in calcvar similar ideas also occur. A related concept might also be that of a "blow up-limit", i.e. if you take the solution to the variable $a$ problem and rescale it around $x_0$, then in the limit will usually be a solution to the constant $a$-problem. (though on $\mathbb{R}^d$, instead of with the boundary data you requested). | |
| May 10, 2021 at 19:05 | answer | added | username | timeline score: 1 | |
| May 10, 2021 at 18:42 | comment | added | Mateusz Kwaśnicki | On the probability side, such "freezing coefficients" approach is known as the parametrix method. But I do not know if it is really related to your question. | |
| May 10, 2021 at 17:47 | comment | added | Willie Wong | If I am parsing things right, for $\gamma > 0$, if $M(p,\gamma)$ is finite this would require that $f(x_0) = f^{\delta}_{x_0}(x_0)$ for every $x_0$ and every compatible $\delta$. (This is just using the inner most $\sup$.) I don't think this can be expected. | |
| May 10, 2021 at 16:37 | history | asked | Kernel | CC BY-SA 4.0 |