I believe I have solved a famous open problem. How do I convince people in the field that I am not a crank? I am interested in the situation where you have a very interesting result. For instance, you have solved a very important open problem. However, you are not known in the field and do not have any remarkable publications. Your supervisor thinks the work is good and you submit the work to a high profile journal, but you get rejected.

The thing is that the contribution is very strong. It breaks what most people believe or what they have already proven: e.g., you solve the P vs. NP problem or any other well known open problem. 

The reviewers strongly reject your work with no justification and they do not state why the result is wrong. Examples of reviewer comments include:


"The proof must be wrong." 
"You cannot achieve such a result." 
"You do not understand well the notion of ..."


My question is what to do in this situation? Where to go? If your advisor accepts the work, but the reviewers from the top journal reject the work without even explaining the mistakes,  what should you do?
Your question has some issues.  Given some of the questions you have asked on other SE sites in the last few days, I have some reservations about whether your question is being asked in good faith, but taken on its own merits it is a reasonable question so I will try to answer it.

The main issue is that, even in asking this relatively simple question, your writing is far from clear.  If you cannot write clearly in this situation, your chances of writing up a difficult piece of mathematics or theoretical computer science are less than good.  For instance:


  His/Her supervisor(s) accept the work and they published it in a highly known journal and they get rejected.


Laying aside issues of subject/verb agreement and consistency of tense, the entire sentence doesn't make sense: you can't publish a paper and get rejected.  


  It breaks what most people believe


I don't know what it means to "break what most people believe".


  or what they have already proven,


What? Are you saying that your proof contradicts other proven results?  Taken literally, that would mean that you have shown mathematics to be inconsistent.  In practice this could only mean that if your result is correct then some previously published work is incorrect.  If that's the case then you need to be very clear about that and explain the flaws in the earlier work.  It distresses me that you don't really seem to believe this but are just throwing it off as loose language. 


  i.e., He/She solves the P vs. NP problem or any other well known open problem.


Solving an open problem would not "break what people have already proven"....that's what it means for the problem to be open.  Also saying "P vs. NP problem or any other well known open problem" is a strange bit of coyness: there is no other problem in theoretical computer science (and very few to none in mathematics as a whole) which is "like" P vs. NP.  So it doesn't make sense to give that as an example.  It's like saying "i.e., he found the Holy Grail or some other famous cup".  

In other questions you have spoken specifically about having a proof of P vs. NP and then upon questioning have retreated from this.  This sort of vacillation about what you have done is a red flag of "crankiness" that will make professionals wary.  


  The reviewers strongly reject his/her work with no justification and they said that the result must be wrong.


Saying that the result must be wrong is not just a justification for rejection, it's the best justification.  No professional reviewer will say something is wrong lightly.  Almost any reviewer who says this will point to at least one specific error.  If they do not, then in practice it almost certainly means that the entire document did not make enough sense to them to be more specific. 


  If your advisor accepts the work, the reviewers reject the work without even explain the mistakes (it is the "best" journal in his/her domain) then what he/she must do?


If you submit a paper to the top journal in your field claiming a solution to the top problem in your field, and your paper does not make sense or does not evince even a correct understanding of the problem, then the editors are likely not to want to spend much time in response.  On the other hand, if you are sincerely interested in getting their expertise, it seems reasonable to write back very politely and ask for more specifics about the error.  If your response is in any way argumentative then you risk the editorial staff thinking that you will keep hounding them ad infinitum, and at some point they have to stop replying.  So you should write back saying that you are not considering resubmitting the paper to that journal but for your own progress it would be extremely helpful to know what is wrong with it.  You could also mention that your supervisor found the paper to be correct.

In fact you could be getting more help on this from your supervisor.  If you have really "solved P vs. NP problem or any other well known open problem" and your supervisor believes your solution to be correct, why isn't your supervisor moving heaven and earth to be sure your work is getting the attention it deserves?  That doesn't add up.  The two possible explanations seem to be (i) your supervisor is being too polite with you: s/he does not actually believe that you have solved P vs. NP; and (ii) your advisor's imprimatur does not carry any weight in the community whatsoever.  The latter unfortunately means his/her opinion on the correctness of your work is not worth very much.  

A good way to find out whether it's (i), (ii) or -- I do admit that anything is possible! perhaps the top journal in your field is unfairly ignoring your revolutionary work -- is to seek your advisor's help in getting another faculty member to evaluate the work, preferably someone in the department that you can speak to recently.

Finally, you seem to have some real worries that if an unknown person solves a famous problem then it somehow doesn't count.  This is really not the way academia works, provided the unknown person is capable of presenting the work in a way which makes sense to the experts (and if not, what a shame, but what else could one possibly expect?).  Have you heard of the recent example of Yitang Zhang?  Zhang was a non-tenure-track lecturer at the University of New Hampshire when he stunned the mathematical world by proving the existence of bounded prime gaps.  He submitted his work to the top mathematical journal...and by all accounts they accepted it with unusual speed.  In other words, they received a paper from someone they had probably never heard of, looked at it quickly and saw that it was a plausible attack on a huge open problem, and as a result they sprung into action much more rapidly and thoroughly than for most submissions they get.  This is an amazing story, but a true one, and it shows how the community responds to a real situation like this.