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I was wandering what i say about product $$ u\eta $$ where $u\in W^{1,2}(\Omega)$ and $\eta\in W^{1,2}_0(\Omega)$. In particular, when i can say that $$ u\eta\in W^{1,2}_0(\Omega). $$ Is it necessary to make more assumptions about u?

Revzora
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  • Look out for Hölder's inequality. As it is you will probably only get $u \eta \in W_0^{1,1}$. For the result you want, you'd need $u \in W^{1,\infty}$. – Cahn Feb 03 '19 at 13:20
  • @Marvin: If you first use Sobolev's embedding theorem, one can show a little bit more than $W^{1,1}_0(\Omega)$. And you do not need $u \in W^{1,\infty}$ since one could also assume more regularity of $\eta$. – gerw Feb 04 '19 at 07:36
  • @gerw True, you are correct that one can show more. And yes, I thought it is only allowed to change the regularity of $u$ in the way the question was proposed. Good answer, +1. – Cahn Feb 04 '19 at 10:19

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For $u \, \eta \in W_0^{1,2}(\Omega)$ you have to prove three things:

  • $u \, \eta \in L^2(\Omega)$,
  • $\nabla( u \, \eta ) \in L^2(\Omega)$, and
  • the trace of $u \, \eta$ is zero.

The first is easy to get and the last one follows basically from the fact that the trace of $\eta$ is zero. The second one is hardest and needs additional assumptions. From the product rule, it is sufficient to check that $u \, \nabla \eta$ and $\eta \, \nabla u$ are in $L^2$. These can be achieved by combining Sobolev's embedding theorem with Hölder's inequality. For example, you it is sufficient to assume one of the following (this list is not exhaustive):

  • $u \in W^{1,\infty}(\Omega)$,
  • $\eta \in W^{1,\infty}(\Omega)$,
  • $u, \eta \in L^\infty(\Omega)$,
  • $u, \eta \in W^{1,p}(\Omega)$ with $p$ large enough (depending on the dimension).
gerw
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  • Thank you for your answer. I know to prove first and second point (with assumptions) but now it is not clear to me how to prove the last point (that is the trace is zero). – Revzora Feb 04 '19 at 14:36
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    You can approximate $\eta$ by compactly supported functions. The product of these functions with $u$ will have zero trace. – gerw Feb 04 '19 at 18:58