Does Heapsort work in time o(n log n) in the best case?
The paper: The Analysis of Heapsort by Schaffer and Sedgewick shows that 


  Theorem 1: Heapsort requires that at least $\frac{1}{2} n \lg n - O(n)$ data moves for any heap composed of distinct keys.


and that


  Theorem 3: The average number of data moves required to Heapsort a random permutation of $n$ distinct keys is $\sim n \lg n$.




It also mentions that 


  If equal keys are allowed, the best case is clearly linear [3]: consider the case of a heap with all keys equal.


The citation to [3] here is TechReport: On Heapsort and its Dependence on Input Data.