This is an extremely broad question. Maybe you should work with browser searches to narrow your focus. You may or may not be interested in application level affinity calls, or in methods external to the application for setting preferences for mapping threads to cores.
In general, there probably isn't very good support for security between multiple applications running on a multi-core system, beyond what is required to make them work reliably, and the same basic stuff you would have on any system. Each core has its own stack and lowest level cache, but higher level cache and main memory will be shared among cores.

Hi,

I agree with Tim the question is very broad. For Cita:

I'm looking for information on running or dedicating applications to running a single core.

you can start to look into the taskset or numactl ulitities that can set application affinities to particular cores and sockets within Linux OS. Separation of applications using virtualization (running each application in its own virtual machine) is often considered as a method to increase security.

Roman