O.K., this is Pli-Grip. It is made out of a thin sheet metal and is made to curve around any shape, usually on the outside back of a piece of upholstered furniture, where it is butted up to a welt and stapled to the wood frame. You push the fabric in under the teeth, and then flatten it down making the teeth grab and hold the fabric. It comes in long rolls and is "L" shaped. Every upholstery shop should have some around.
For this job you have to bend it so it is flat. You simply cut the shape of the door opening and staple the Pli-Grip teeth side down to the PVC foamboard. It should be thicker than what is used for a panel to hold its shape better and have enough thickness to staple to, usually 1/4" thick. You can also cut the shape out of plywood, or if you have metal working skills, cut it out of light steel and punch diamond shapes all around the edge of the shape you cut, in which case you don't need the Pli-Grip. This is then screwed to the area above the door opening. When you install your headliner, the excess fabric is trimmed off and pushed up under the shaped piece and under the teeth of the Pli-Grip. The teeth will hold the headliner in place. When you have the headliner fabric in place the way you want it, take a rubber mallet or dead blow hammer and tap the strip down to engage the teeth even more.
No one lives forever, the trick is creating something that will.