Regarding Trace of a function in $W^{1,p}$

Question

By Trace theorem it implies that if $u\in W^{1,p}(\Omega)$ then $\text{Tr}(u)\in L^p(\partial \Omega)$. Now, if $\text{Tr}(u)\in L^p(\partial \Omega)$ can it be said that $u\in W^{1,p}(\Omega)$. Are there any results for this?

Answer 1

As mentioned by Artic Char, this cannot be true.

However, maybe you really want to know the following: Given $v \in L^p(\partial\Omega)$, does there exist $u \in W^{1,p}(\Omega)$ with $\operatorname{Tr}(u) = v$.

This is not true, since the trace of $u \in W^{1,p}(\Omega)$ has higher regularity then $L^p(\partial\Omega)$. In fact, one can show that the trace belongs to the so-called trace space $W^{1-1/p,p}(\partial\Omega)$. Then, the trace operator $\operatorname{Tr} \colon W^{1,p}(\Omega) \to W^{1-1/p,p}(\partial\Omega)$ becomes surjective.