Timeline for Different ways to prove $L^p$-estimates for the heat equation
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Dec 23, 2022 at 23:01 | comment | added | Giorgio Metafune | It is as for elliptic equations. The proof of $L^p$ estimates for the Laplacian is the same as for boundedness of Calderon-Zygmund operators. For general second order operators, $L^p$ estimates are usually deduced by the laplacian "freezing" the coefficients, that is by perturbation and this last step is quite a standard machinery. | |
| Dec 23, 2022 at 16:40 | comment | added | Ayman Moussa | Thanks @GiorgioMetafune. Since you seem rather informed on the subject: I am really surprised that for the specific case of the heat equation there's no other method than those "general" approach which in fact cover a lot more. I would expect that the structure of the equation itself could help to simplify the proof. Any comment in that direction is also welcome ! | |
| Dec 23, 2022 at 15:20 | comment | added | Giorgio Metafune | The more recent research concerning boundedness of singluar integrals in homogenuous spaces simplyfieses both exposition and proofs, in my opinion, even though it could appear more abstract. | |
| Dec 23, 2022 at 15:19 | comment | added | Giorgio Metafune | The book of Liebermann is complete but difficult to read, unfortunately but the approaches in the answer above are much more readable. The main point is to prove $L^p$ estimates in the whole space for $D_t-\Delta$ (no boundary conditions, no intial data). The rest is just perturbation as far as we confine to uniformly continuous top order coefficients. For $D_t-\Delta$ either one uses the kernel, or multipliersand in any case some harmonic analysis. | |
| Dec 23, 2022 at 13:20 | history | answered | Ayman Moussa | CC BY-SA 4.0 |