Timeline for Regularity up to boundary of a solution $u\in W^{1,p}\cap W^{2,2}(\Omega;\Bbb R^m)$ to $\Delta^2 u = -\text{div}\, F$
Current License: CC BY-SA 4.0
5 events
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| Jul 2, 2021 at 13:03 | comment | added | Leo Moos | I'm a bit surprised this can't be found in the literature on biharmonic maps; perhaps you could explain where the literature falls short? For example, concerning interior regularity there is a paper of Chang-Wang-Yang entitled 'A regularity theory for biharmonic maps' that states the bound $\lvert w \rvert_{3,q} \leq C \lvert F \rvert_{0,q}$, at least when $w = 0$ and $\partial_\nu w = 0$ on the boundary. | |
| Jun 29, 2021 at 3:24 | history | edited | BigbearZzz | CC BY-SA 4.0 |
added 115 characters in body
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| Jun 29, 2021 at 3:21 | comment | added | BigbearZzz | @LeoMoos You are right, that was a bad example. Let me amend that a bit to reflect what I actually wanted to say. | |
| Jun 28, 2021 at 12:27 | comment | added | Leo Moos | This is in regards to the example that you mention after your question. Perhaps I'm off the mark, but isn't it illusory to aim for $u$ more regular than its proposed boundary values? The Sobolev inequalities don't allow embedding $W^{2,2}$ inside $C^1$ in general. | |
| Jun 21, 2021 at 13:48 | history | asked | BigbearZzz | CC BY-SA 4.0 |