Cores for parabolic operators with unbounded coefficients

Abstract

Let A = ∑ i , j = 1 N q i j ( s , x ) D i j + ∑ i = 1 N b i ( s , x ) D i be a family of elliptic differential operators with unbounded coefficients defined in R N + 1 . In [M. Kunze, L. Lorenzi, A. Lunardi, Nonautonomous Kolmogorov parabolic equations with unbounded coefficients, Trans. Amer. Math. Soc., in press], under suitable assumptions, it has been proved that the operator G : = A − D s generates a semigroup of positive contractions ( T p ( t ) ) in L p ( R N + 1 , ν ) for every 1 ⩽ p < + ∞ , where ν is an infinitesimally invariant measure of ( T p ( t ) ) . Here, under some additional conditions on the growth of the coefficients of A , which cover also some growths with an exponential rate at ∞, we provide two different cores for the infinitesimal generator G p of ( T p ( t ) ) in L p ( R N + 1 , ν ) for p ∈ [ 1 , + ∞ ) , and we also give a partial characterization of D ( G p ) . Finally, we extend the results so far obtained to the case when the coefficients of the operator A are T-periodic with respect to the variable s for some T > 0 .