Let $\Omega$ be an open set with boundary such that $\partial\Omega = \Gamma_1 \cup \Gamma_2$ where $\Gamma_1$ and $\Gamma_2$ are closed and smooth but disconnected, in particular think of $\Omega$ to be an annulus.
I'm looking for existence and regularity results (in Sobolev or Sobolev-Bochner spaces) for parabolic equations of the form
$$u_t - \Delta u = f$$ $$u|_{\Gamma_1} = g$$ $$u|_{\Gamma_2} = h$$
and what kind of regularity is needed for $f, g, h$. Does anyone have a reference?