As Giuseppe correctly notes, DR is per shared (i.e. broadcast) LAN segment.
However, did want to add, you can set priorities for router DR elections.
For example, given:
Both shared segments would have a DR and BDR, but R1 and R2 would be the candidate routers for shared1 while R2 and R3 would be the candidate routers for shared2. I.e. you might have R1 and R3 as DRs, or R2 (for shared1) and R3 as DRs, or R1 and R2 (for shared2) as DRs, or R2 as DR (for both shared1 and shared2).