# Preprint of the project: SFB 701: Spectral Structures and Topological Methods in Mathematics - Project B3

## Numerical Analysis of equivariant evolution equations

15-042 Denny Otten.
A new $$L^p$$-Antieigenvalue Condition for Ornstein-Uhlenbeck Operators

In this paper we study perturbed Ornstein-Uhlenbeck operators $\left[\mathcal{L}_{\infty} v\right](x)=A\triangle v(x) + \left\langle Sx,\nabla v(x)\right\rangle-B v(x),\,x\in\mathbb{R}^d,\,d\geq 2,$ for simultaneously diagonalizable matrices $$A,B\in\mathbb{C}^{N,N}$$. The unbounded drift term is defined by a skew-symmetric matrix $$S\in\mathbb{R}^{d,d}$$. Differential operators of this form appear when investigating rotating waves in time-dependent reaction diffusion systems. As shown in a companion paper, one key assumption to prove resolvent estimates of $$\mathcal{L}_{\infty}$$ in $$L^p(\mathbb{R}^d,\mathbb{C}^N)$$, $$1\lt p \lt \infty$$, is the following $$L^p$$-dissipativity condition $|z|^2\text{Re}\left\langle w,Aw\right\rangle + (p-2)\text{Re}\left\langle w,z\right\rangle\text{Re}\left\langle z,Aw\right\rangle\geq \gamma_A |z|^2|w|^2\;\forall\,z,w\in\mathbb{C}^N$ for some $$\gamma_A>0$$. We prove that the $$L^p$$-dissipativity condition is equivalent to a new $$L^p$$-antieigenvalue condition $A\text{ invertible}\quad\text{and}\quad\mu_1(A)>\frac{|p-2|}{p},\,1\lt p \lt\infty,\,\mu_1(A)\text{ 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 $$L^p$$-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.