So here is my situation: I am applying for a PhD for Fall 2022. Not too long I found ago that my ex boyfriend had used my name online to post all kinds of crap. This includes revenge porn, as well as racist and anti-semitic comments. I have filed police reports, but all of this is still attached to my name (which is a very rare name btw) and available online, and probably will never go away. I am so scared of not being admitted because of this. Is there anything I can do?
The pictures and the posts are rather different issues. Whether there are nude pictures of an applicant available on the internet is no business of an admissions committee, regardless of how they got there. This aspect is covered well in another question:
For the racist comments however, the fact that you did not write them is essential. If you had written them, it would be very legitimate to hold it against you. If these show up on the first page of Google results for your name, you may want to include a statement with your applications that horrific comments with your name attached to them exist on the internet, and that you neither wrote them nor agreed with them.
I don't entirely agree that this is a duplicate.
I would put a note somewhere in your application materials that you have been a victim of online abuse and that online information about you is likely invalid. You don't need to give details. People understand the persistence of such things.
If anyone asks about it, say in an interview, just refer them too the police department to which you made complaint.
No one should hold you responsible for the acts of others, especially an abusive former partner. But it is better if others give the defense than if you try to explain it yourself.
If any of your recommenders are aware of the situation, perhaps they can make a similar statement, without details. And especially, speak to your lack of racist actions or attitudes.