Timeline for duality of sobolev spaces. Representation of elements in the dual
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 5 at 13:43 | comment | added | Hannes | Well, yes, if you want to do this extremely rigorously coming from a distributional point of view; the point being that $\langle \operatorname{div} F,\varphi\rangle$ can be estimated by $\lVert F\rVert_{L^q(\Omega)^n} \lVert \nabla\varphi\rVert_{L^p(\Omega)^n}$, so you extend from test functions to $W^{1,p}_0(\Omega)$ by density. (Alternatively just define the $\operatorname{div}$ directly on $W^{1,p}_0(\Omega)$ as in the second part of (2) in OP and convince yourself that it gives you a bounded linear operator..) | |
| Dec 4 at 21:12 | comment | added | Alucard-o Ming | @Hannes I know that, this is exactly Step 1. But $f-divF$ makes sense only when acting on test functions,not on $W^{1,p}$ function , so then we must apply $f-divF$ to a sequence of test functions conversion in $W^{1,p}$ right ? | |
| Dec 4 at 20:41 | comment | added | Hannes | It is well understood how to characterize elements of dual spaces of the Sobolev spaces $W^{1,p}_0(\Omega)$, see for examples Proposition 9.20 in the book of Brezis. Essentially, one can represent it by $f - \operatorname{div} F$ with $(f,F) \in L^q(\Omega)^{1+n}$ and $q$ the conjugate exponent of $p$, as hinted at by Pietro. | |
| Dec 4 at 20:07 | comment | added | Alucard-o Ming | @PietroMajer sorry can you clarify ? the problem is that the representation si not unique ? but is (1) a correct representation ? | |
| Dec 4 at 19:16 | comment | added | Pietro Majer | Embed $W_0^{1,p}(\Omega)$ isometrically onto a subspace of $L^p(\Omega)^{n+1}$ via $f\mapsto (f,\partial_1f,\dots,\partial_nf)$, and use Hahn-Banach. This yields to a (not unique) representation for every linear form by $L^q(\Omega)^{n+1}$. | |
| Dec 4 at 18:58 | history | asked | Alucard-o Ming | CC BY-SA 4.0 |