Yagami Light and L VS Joker (The Dark Knight)

Scenario 1: Who can catch who first?

The Joker is in Japan and L and Light are working together to stop him. L and Light start out knowing nothing about the Joker besides what he's called and what he might look like, ditto for the Joker. L has all of the Japanese police working with him to catch the Joker, and Light has the Death Note and Ryuk at his disposal. With that said, the Joker has already formed a small group underneath him and all of the Japanese mobs are dead scared of him, so the Joker can use any weapon that the Japanese mob can give him.  

So who can catch who dead or alive first?

Scenario 2: Straight on fight.

L, Light, and the Joker are all alone in an abandon ware house. L has nothing on him, Light has the Death Note, and the Joker has a knife and a pencil. Who would win in a fight to the death? Oh and no prep time for both sides.

This is an odd fight.

L solos