Say you, in the middle of a battle, cast a spell such as Magnificent Mansion. You go inside and then start casting Conjure Elemental. You then order the elemental to exit the mansion and start fighting something you saw on the outside. Can you maintain concentration on the elemental if you are in a separate plane than the elemental?
Yes, Concentration can be kept across dimensions
The PHB (p. 203) lists what can break concentration: casting another concentration spell, taking damage, or being incapacitated/killed.
in addition it mentions
The DM might also decide that certain environmental phenomena, such as a wave crashing over you while you’re on a storm-tossed ship, require you to succeed on a DC 10 Constitution saving throw to maintain concentration on a spell.
In the description of what events might break concentration the example is something that we would expect to make someone lose focus in a real world sense. If crossing dimensions was a turbulent affair your DM might ask for a check but the description for Magnificent Mansion makes it seem that entering/exiting is no more disruptive than entering a normal building. So if you wanted to spend 10 turns in combat casting Magnificent Mansion and then another 10 turns casting Conjure Elemental and then command it to go outside and fight you could certainly try.
It should be noted that some spells, such as Planar Binding and Sending, specify in their text that being on a different plane alters their effects.
This question might also provide some insight into a different example of concentration across dimensions