A new Lp-antieigenvalue condition for Ornstein–Uhlenbeck operators

Abstract

In this paper we study perturbed Ornstein–Uhlenbeck operators[L∞v](x)=A△v(x)+〈Sx,∇v(x)〉−Bv(x),x∈Rd,d⩾2, for simultaneously diagonalizable matrices A,B∈CN,N. The unbounded drift term is defined by a skew-symmetric matrix S∈Rd,d. Differential operators of this form appear investigating rotating waves in time-dependent reaction diffusion systems. As shown in a companion paper, one key assumption to prove resolvent estimates of L∞ in Lp(Rd,CN), 1<p<∞, is the following Lp-dissipativity condition|z|2Re〈w,Aw〉+(p−2)Re〈w,z〉Re〈z,Aw〉⩾γA|z|2|w|2∀z,w∈CN for some γA>0. We prove that the Lp-dissipativity condition is equivalent to a new Lp-antieigenvalue conditionA invertibleandμ1(A)>|p−2|p,1<p<∞,μ1(A) first antieigenvalue of A, which is a lower p-dependent bound of the first antieigenvalue of the diffusion matrix A. This relation provides a complete algebraic characterization and a geometric meaning of Lp-dissipativity for complex-valued Ornstein–Uhlenbeck operators in terms of the antieigenvalues of A. The proof is based on the method of Lagrange multipliers. We also discuss several special cases in which the first antieigenvalue can be given explicitly.