To fix this problem, you can easily adjust the hinges on your cabinet doors. By adjusting the hinges, you can eliminate the gap under the doors and create a smooth surface that will allow the doors to close properly.
This is a common problem and it's usually caused by the hinge pin being inserted too far into the hinge. The pin should protrude from the hinge about 1/8 to 1/4 of an inch. If it's too far in, you'll have to remove it, put it in the proper position, and then reinsert it.
Dip wooden toothpicks into wood glue and then stuff the cabinet hinge hole with as many toothpicks as will fit. Break off the toothpicks so they are flush with the cabinet. Once the glue is dry, reattach and screw the hardware back into place.