Does there exist a non zero function $u\in C(\mathbb{C})$, harmonic in $\mathbb{C}\setminus\mathbb{T}$ that satisfies the following equation: $$u(z)+u(-z-2)=0\:\:\forall z\in\overline{\mathbb{D}}$$
Where $\mathbb{D}=\{z: |z|<1\}$ and $\mathbb{T}=\{z: |z|=1\}$.