PBKDF2 with SHA-256/512 for low entropy input? Is there any reason to use something stronger than SHA-1 with PBKDF2 if my input is fairly low entropy?  I.e. if the passwords being hashed are only about 30-bits of entropy.  

I can't find it now, but someone stated if you want to preserve the entropy of high entropy intput, then use PBKDF2 with a higher SHA.  So I'm wondering if the inverse is true, that there is no real gain to using SHA-512 if your input is low entropy.

I've seen one case mention that using SHA-512 would increase the memory requirements for attackers.  In some cases, I know that increasing memory requirements will make GPU hashing less effective.  Not sure if that's true in this case.  https://cryptosense.com/blog/parameter-choice-for-pbkdf2/
There is not much of a substantive difference between using SHA-1 or SHA-256 with PBKDF2 to hash a password: they use nearly the same amount of memory (20 bytes vs. 32 bytes of state, plus 8 bytes of message length and 64 bytes of input buffer), and SHA-512 uses only double what SHA-256 uses (64 bytes state, 16 bytes length, 128 bytes buffer).

If you can, forget PBKDF2 and consider using Argon2id instead, so that if you have (say) 128 MB free for password hashing—and perhaps a few extra CPU cores—you can use it all to drive up an adversary's costs.