Sounds viral or possibly fungal. What are you doing for your immune system to boost it? It doesn't sound to me like you have any medical options since the doctors don't even recognize it. But most skin problems are the result of the body trying to get rid of something, to detoxify itself. I work extensively with immune system support, and all of the many skin issues (eczema, psoriasis, dermatitis, shingles, hives and so on) respond to the same regimen of providing additional nutrients to the body to help it process normal allergens or triggers, so they are eliminated harmlessly or just not reacted to at all. Some people get results in a few weeks, some in several months - it's individual and some people definitely have more stubborn problems. But I've never seen anyone NOT get results if they are consistent. There is some astounding new research on a natural peptide which is really nature's superfood, the first dietary ingredient known to affect gene "expression" (which means the instructions to each cell to tell it which genes to turn on and off, which is what determines proper cell functioning).
Yours don't sound like they are totally hormone dependent, because your daughter and your son both get them. So that tells me these are epigenetic in nature, which we've learned can be hereditary but is reversible. Kids usually respond sooner, but there's a very easy and totally safe way to address these epigenetic changes, which is great for prevention as well as dealing with an existing problem. It puts your body back to where it's supposed to be before all the lifestyle changes kick in (which can be our own choices but also environmental or food triggers).