I need to solve the heat equation without sources (Laplace’s equation) on the green domain which is bounded by two rectangles shown below:
Is it possible to do that analytically? So far I haven’t been able to find any reference that it is not possible, and from my math classes in college (some years ago) I remember that through a conformal mapping one could change the solution domain to a domain that is easier to solve.
No, this kind of domain calls for numerical solutions. In principle, there is a version of Schwarz-Christoffel formula for conformal map of a doubly-connected polygonal domain onto a circular annulus (details and references here), but one usually can neither find the constants it involves analytically, nor perform the integration it requires.
And even if one can work out the conformal map, the circular annulus doesn't admit a simple Poisson kernel the way the disk does; there is an infinite series representation of it, see section 5.
Taking this really complicated series, composing it with a conformal map that one can't really evaluate, and then integrating against boundary data analytically, is several steps beyond what's feasible.