I have corian counter tops in my kitchen and also a corian sink. I put a heavy object
into the sink and I knocked a 4inch x 12inch piece out of the bottom. It is out of warranty.
Can the piece be glued or somehow put back into place where it will hold water and look
OK?

Yes,keep the pieces from any more damage.

The same process used to join counter top extensions and build-up ,back splashes,etc can be used for this repair.

Contact a kitchen counter top company or Corian,or Dupont.

Hope you succeed and good luck,

KBC