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-



Continuity of an extension map
Yes, this extension works. Your extended function is clearly in $H^1$ on the annulus as well as
in $R^n\backslash B\_2$. Since the traces on $\partial B\_2$ agree, it is then in $H^1$ on $R^n\backslash B\_r$.


