Availability / Capacity / Incident / Problem ManagI received folement - relation

Answer:
Let's clarify Availability Management (AM). You will perform activities in scope of AM during the design of the service as well as in scope of operational activities once service is in live environment (like you mentioned - Incident/Problem Management). This article will help you understand AM:
Availability Management – calculating for improvement https://advisera.com/20000academy/blog/2013/08/21/availability-management-calculating-improvement/
Availability Plan is one of the products of the AM, and you can find more info here
ITIL Availability Plan – A document you need, but probably don’t have https://advisera.com/20000academy/blog/2015/05/19/itil-availability-plan-a-document-you-need-but-probably-dont-have/

Capacity Management is rela ted to the performance of the service (and it's constitutional components i.e. service assets). This articles will help you with CM:
ITIL and ISO 20000 – How to setup the Capacity Management process https://advisera.com/20000academy/blog/2016/02/16/itil-and-iso-20000-how-to-setup-the-capacity-management-process/
Three faces of Capacity Management https://advisera.com/20000academy/knowledgebase/three-faces-capacity-management/
ITIL Reactive and Proactive Capacity Management https://advisera.com/20000academy/blog/2015/04/07/itil-reactive-and-proactive-capacity-management/

Further on, once service is in live environment, there could be issues which are related to capacity or availability. Therefore, AM and CM are involved (usually as 2nd or 3rd level support) in resolving incidents i.e. problems.