Rabinowitz minimal periodic solution conjecture
In 1978, Professor Paul Rabinowitz proved the existence of T-periodic solutions for autonomous Hamiltonian systems whose Hamiltonian function is super-quadratic at infinity and zero for any given T>0 by using a minimax variational method. Because the minimal period of this solution could be T/k for some positive integer k, he asked whether such a system possesses always a T-periodic solution with T as its minimal period. This is the famous Rabinowitz minimal periodic solution conjecture. In the last more than 40 years, many mathematicians have studied this conjecture and got many interesting results. But the conjecture is still open under the original conditions of Rabinowitz. In this lecture, I shall give a brief survey on the main results obtained and methods used in these studies so far, and hope to lead to more interests on this conjecture.