On principal eigenvalues for periodic parabolic problems with optimal condition on the weight function

Let fl be a bounded domain in R-N and let in be a T-periodic function such that its restriction to Omega x (0, T) belongs to L-s((0, T), L-upsilon(Omega)) for some upsilon > N/2 and s > 2 upsilon /2 upsilon -N with upsilon > 1 and s > 2. We give necessary and sufficient conditions on in for the existence, uniqueness, and simplicity of the principal eigenvalue for the Dirichlet periodic parabolic eigenvalue problem with weight in. (C) 2001 Academic Press.