You need to search on "foundation" and "engineering".
Several years ago I worked for a company who did this type of work in the Boston area. Basically, we would jack up the foundation, build a new foundation and then lower the existing structure on to the new foundation.
However, it not as simple as that. The firm was run by engineers who studied the structure and soil conditions to ensure the new foundation woud be able to handle the load.
Boston is famous for sinking foundations since a good part of the city sits on sand that was brought in and filled up the harbor. So, you should be able to find some firms that specialize in this type of work.