Let $X$ be a smooth projective variety (say, over a field of characteristic zero). Let us say that strong Kodaira vanishing holds for $X$ if $$ H^q(X,\Omega^p\otimes L)=0 $$ for every $p\geq 0$, $q>0$ and an ample line bundle $L$ on $X$.

My questions are now these:

1) Does strong Kodaira vanishing hold for $X={\mathbb P}^N$?

2) Does it hold for partial flag varieties of a semi-simple group $G$?

3) What tools are there for proving that strong Kodaira vanishing holds for a given variety $X$?