Suppose $\delta\in (0,1)$ and $r<1+\delta.$ Suppose moreover we are given a sequence of functions $u_m\in H^{1/2,2}(\partial B_r(0))$, where $B_r(0)$ denotes the euclidean $n-$dimensional ball. Assume that $u_m\to 0$ strongly in $H^{1/2,2}(\partial B_r(0))$. Then I would like to extend these functions to functions in $H^{1,2}(\mathbb R^n\setminus B_r(0))$ by means of a function $$\eta:H^{1/2,2}(\partial B_r(0))\to H^{1,2}(\mathbb R^n\setminus B_r(0)),$$ defined as follows: setting $v=\eta(u)$ we would have

$v=0$ in $\mathbb R^n\setminus B_2(0),$

$v=u$ on $\partial B_r(0)$,

$-\Delta v=0$ in the annulus $B_2(0)\setminus B_r(0)$.

Is this possible ? Moreover, if such a map existed, would it have some continuity properties? I cannot see if my question is just a consequence of the usual extension theorem because I'm requiring harmonicity in an annulus. Even references are welcomed since I was not able to find any of them.

Best Regards.

-Guido-