Polysporin does work good for chapped lips since it helps to seal moisture into your lips and protect them from drying out.
You can also use neosporin on chapped lips or even Blistex Lip Med which all work great at treating chapped lips and keeps them moisturized.
Another thing you can do to heal chapped lips and prevent chapped lips is to gently rub your lips with a homemade mixture of olive or coconut oil and sugar and wipe clean with a face washer or a splash of warm water.
That will help to exfoliate your lips and help seal moisture into your lips naturally.
Wearing lip balm regularly and doing the above will help with chapped and dry lips.