My daughter gave me an early Christmas gift of a book from my Amazon Wish List. She bought it locally, and this was the only copy the store had available. Notice how badly the cover is curled. The fiber in the paper has been broken, too. So, just putting weight on it and waiting will not help.
Step 1: Idea
I decided to add something to the back side that would pull the cover back into place. I needed a curved form to do it. I do not know what this is called, but my wife puts bread dough for long loaves onto each half when she bakes certain kinds of bread.
Step 2: Add a reverse curve
Here I am holding the bend in the cover against the opposite curve of the loaf holder. It looks awkward here because I had to hold the camera with one hand, too.
Step 3: Reinforce the backside
I used a good quality cellophane packing tape about 2 inches wide. Although difficult to see in the photo, the yellow corner markers show the location of the packing tape.
While holding the book's cover in a curve the reverse of the curve from shelf wear, I applied a piece of tape beginning at the edge of the cover and rolling it down smoothly to avoid creases as much as possible.
When finished rolling out the tape with my fingers I trimmed any tape overhang with a good pair of scissors.
Step 4: The result
All in all, it turned out well. Compare this with the first photo. As you might guess, the front cover needed the same reinforcement.