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.
