Product from function in $W^{1,2}_0(\Omega)$ and function in $W^{1,2}(\Omega)$

Question

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?

Answer 1

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).