This answer is coming very late, but may be useful to people who land up on this thread. By referring to different departments, I mean different PhD programmes.
Regarding the PhD application:
Since the same professor is in all the three departments and he is conducting research which are aligned with your research goals, there is no harm in applying to all 3 programs. It may make a stronger case for your selection as well (not a guarantee always, but no negative impact surely). The graduate committee is made of professors who are very well aware of the fact that almost all phd applicants apply to multiple phd programmes, the difference here is that they know where 2 other applications of yours went. They may ask you to set a priority, that's all. Showing interest is different from receiving an offer, professors may be interested in multiple candidates, but due to constraints give out offers to a smaller pool. They should not hold anything against you. If they do, you are better off not joining that programme.
I applied to 3 phd programmes in the same university, and got selected in my second choice. All 3 departments knew that I had applied to the other 2 as well. They had no issues with it.
Often different departments in the same university have different ways of selecting candidates and providing funding. In my case, two of the departments gave out department-centralized PhD offers, where we would have to choose our advisor after joining the programme. However in the third department at the same university, funding was through the grants of individual faculty. So the PhD advisor was fixed from the beginning, and the funding was specific to that project's requirements. You would have to ask the graduate programme manager for exact details.