when looking at the photos, the obvious answer is to add concrete,however, I think the new concrete will bust when the carpet crew put in their tack strips. Need to hear how to fix it right. It is happening at both exterior doors.
Is that a wood plate under the door?
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Remodeling Contractor just outside the Glass City.
Cut or chisel out a channel in the concrete and insert a PT wood filler strip to nail the tackless into.
You could drill and secure your own tack strips with concrete screws, and fill the spalling with PL Premium to help keep the bond.
Fix it with an epoxy patch compound. And try to convince the carpet crew to glue down their tack strip.
You don't want a piece of wood there as it would be a magnet for rot.
The other option is to not use carpet up against the door, but have a vinyl or tile skirt. In our part of the country you never see carpet up against an exterior door.
Of all the preposterous assumptions of humanity over humanity, nothing exceeds most of the criticisms made on the habits of the poor by the well-housed, well-warmed, and well-fed. --Herman Melville
I am not sure, but I think I see signs of water penetration under those doors. If So, You don't want carpet there until you pull the doors and re-set them, correcting the problem so that water does not ruin the carpet
We did the best we could...