The identification problem for complex-valued Ornstein–Uhlenbeck operators in L p ( R d , C N )

• Published in: Semigroup forum 2017-08, Vol.95 (1), p.13-50
• Main Author: Otten, Denny
• Format: Article
• Language: English
• Subjects: Differential equations; Dissipation; Ingredients; Mathematical analysis; Matrices (mathematics); Matrix methods; Operators; Time dependence
• ISSN: 0037-1912; 1432-2137
• DOI: 10.1007/s00233-016-9804-y

In this paper we study perturbed Ornstein–Uhlenbeck operators L ∞ v ( x ) = A ▵ v ( x ) + S x , ∇ v ( x ) - B v ( x ) , x ∈ R d , d ⩾ 2 , for simultaneously diagonalizable matrices A , B ∈ C N , N . The unbounded drift term is defined by a skew-symmetric matrix S ∈ R d , d . Differential operators of this form appear when investigating rotating waves in time-dependent reaction diffusion systems. We prove under certain conditions that the maximal domain D ( A p ) of the generator A p belonging to the Ornstein–Uhlenbeck semigroup coincides with the domain of L ∞ in L p ( R d , C N ) given by D loc p ( L 0 ) = v ∈ W loc 2 , p ∩ L p ∣ A ▵ v + S · , ∇ v ∈ L p , 1 < p < ∞ . One key assumption is a new L p -dissipativity condition | z | 2 Re w , A w + ( p - 2 ) Re w , z Re z , A w ⩾ γ A | z | 2 | w | 2 ∀ z , w ∈ C N for some γ A > 0 . The proof utilizes the following ingredients. First we show the closedness of L ∞ in L p and derive L p -resolvent estimates for L ∞ . Then we prove that the Schwartz space is a core of A p and apply an L p -solvability result of the resolvent equation for A p . In addition, we derive W 1 , p -resolvent estimates. Our results may be considered as extensions of earlier works by Metafune, Pallara and Vespri to the vector-valued complex case.