Let $B_r:= \{ x \in \mathbb{R}^N; |x| < r\}$. Consider $u \in H^{1}_{0}(B_1)$ and $f \in L^2({B_1})$ such that
$$\int_{B_{\frac{1}{2}}} \nabla u \nabla v dx = \int_{B_{\frac{1}{2}}} fv dx, \forall v \in H^{1}_{0}(B_{\frac{1}{2}})$$ and $$\int_{A} \nabla u \nabla v dx = \int_{A} fv dx, \forall v \in H^{1}_{0}(A)$$ where $A$ denotes the interior of the set $B_1 - B_{\frac{1}{2}}$
With such information it is possible to affirm that (in the weak sense) $- \Delta u =f $ in $B_1$ and $u = 0$ on $\partial B_1$?
Thanks in advance