Let $\omega,\Omega\subset\mathbb R^n$, $n\geq2$, be bounded smooth domains so that $\bar\omega\subset\Omega$. Let $1<p<\infty$. Define the boundary space $B=W^{1,p}(\omega)/W^{1,p}_0(\omega)$; this is the space of possible boundary values of a function in $W^{1,p}(\omega)$ in the Sobolev sense.

We can restrict any $u\in W^{1,p}(\Omega)$ to $\partial\omega$ and get a function in $B$. (More formally we can first restrict the function to $\omega$ and the quotient by $W^{1,p}_0(\omega)$.)

Is the set $ A=\{u|_{\partial\omega};u\in W^{1,p}(\Omega),u\text{ is }p\text{-harmonic}\} $ dense in $B$?

My final goal is slightly more complicated, but I feel that I must first understand this problem before I can introduce more moving parts. I am mainly interested in the nonlinear case $p\neq2$, but also references or ideas for the case $p=2$ are also appreciated. I convinced myself that $A$ is dense in $B$ if $p=2$ and the domains are cocentric balls, but this is far from what I eventually need.

A function $u$ is $p$-harmonic if $\Delta_pu:=\operatorname{div}(|\nabla u|^{p-2}\nabla u)=0$ in the weak sense.

Let $\omega,\Omega\subset\mathbb R^n$, $n\geq2$, be bounded smooth domains so that $\bar\omega\subset\Omega$. Let $1<p<\infty$. Define the boundary space $B=W^{1,p}(\omega)/W^{1,p}_0(\omega)$; this is the space of possible boundary values of a function in $W^{1,p}(\omega)$ in the Sobolev sense.

We can restrict any $u\in W^{1,p}(\Omega)$ to $\partial\omega$ and get a function in $B$. (More formally, we can first restrict the function to $\omega$ and then quotient by $W^{1,p}_0(\omega)$.)

Is the set $ A=\{u|_{\partial\omega};u\in W^{1,p}(\Omega),u\text{ is }p\text{-harmonic}\} $ dense in $B$?

I am mainly interested in the nonlinear case $p\neq2$, but references or ideas for the case $p=2$ are also appreciated. I convinced myself that $A$ is dense in $B$ if $p=2$ and the domains are cocentric balls, but this is far from what I want.

A function $u$ is $p$-harmonic if it solves $\Delta_pu:=\operatorname{div}(|\nabla u|^{p-2}\nabla u)=0$ in the weak sense.