Timeline for Duality argument for elliptic regularity
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Jan 14, 2021 at 15:59 | comment | added | user171871 | Yes. Anyway "a trick " with introducing of $D_p$ was a big lesson to me. | |
| Jan 14, 2021 at 11:59 | comment | added | Hannes | @Logiyko So, an estimate as I wrote in the previous comment? | |
| Jan 14, 2021 at 8:51 | comment | added | user171871 | All I need is the existence of a constant in a priori estimate of solution u in a Sobolev space from the Laplacian u in corresponding dual space. | |
| Jan 14, 2021 at 8:41 | comment | added | Hannes | @Logiyko I am not quite sure if this is what you mean, but in the situation mentioned, you do have a constant $C>0$ such that $\|u\|_{W^1_p} \leq C \|\Delta u\|_{(W^1_{p'})^*}$, yes. | |
| Jan 14, 2021 at 8:22 | comment | added | user171871 | So if we need to estimate u in W1,p the first two paragraphs of your answer is enough, isn't it? | |
| Jan 13, 2021 at 15:49 | comment | added | Hannes | @Logiyko Uniqueness is implicit here because the Laplacian on $(W^1_{p'})^*$ is inherited by the one on $(W^1_2)^*$. Check the formulation in Theorem 1.1. in Monique Dauges paper, it is an assertion about the variational $W^1_2$ solution. I have added a paragraph in my answer. | |
| Jan 13, 2021 at 15:48 | history | edited | Hannes | CC BY-SA 4.0 |
Elaborated on the "definition" part
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| Jan 13, 2021 at 15:05 | comment | added | user171871 | As far as I understand the regularity property doesn't imply uniqueness. Moreover in Corollary 3.11 doesn't give a priori estimate of solution with specific "C". So the weak Laplacian doesn't have a continuous linear inverse yet. Which reference can you offer to study "the definition of the Laplacian... by restriction from... "? | |
| Jan 13, 2021 at 13:25 | history | answered | Hannes | CC BY-SA 4.0 |