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Semigroup Forum

, Volume 95, Issue 1, pp 13–50 | Cite as

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

  • Denny Otten
Research Article

Abstract

In this paper we study perturbed Ornstein–Uhlenbeck operators
$$\begin{aligned} \left[ \mathcal {L}_{\infty } v\right] (x)=A\triangle v(x) + \left\langle Sx,\nabla v(x)\right\rangle -B v(x),\,x\in \mathbb {R}^d,\,d\geqslant 2, \end{aligned}$$
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. We prove under certain conditions that the maximal domain \(\mathcal {D}(A_p)\) of the generator \(A_p\) belonging to the Ornstein–Uhlenbeck semigroup coincides with the domain of \(\mathcal {L}_{\infty }\) in \(L^p(\mathbb {R}^d,\mathbb {C}^N)\) given by
$$\begin{aligned} \mathcal {D}^p_{\mathrm {loc}}(\mathcal {L}_0)=\left\{ v\in W^{2,p}_{\mathrm {loc}}\cap L^p\mid A\triangle v + \left\langle S\cdot ,\nabla v\right\rangle \in L^p\right\} ,\,1<p<\infty . \end{aligned}$$
One key assumption is a new \(L^p\)-dissipativity condition
$$\begin{aligned} |z|^2\mathrm {Re}\,\left\langle w,Aw\right\rangle + (p-2)\mathrm {Re}\,\left\langle w,z\right\rangle \mathrm {Re}\,\left\langle z,Aw\right\rangle \geqslant \gamma _A |z|^2|w|^2\;\forall \,z,w\in \mathbb {C}^N \end{aligned}$$
for some \(\gamma _A>0\). The proof utilizes the following ingredients. First we show the closedness of \(\mathcal {L}_{\infty }\) in \(L^p\) and derive \(L^p\)-resolvent estimates for \(\mathcal {L}_{\infty }\). 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.

Keywords

Complex-valued Ornstein–Uhlenbeck operator Identification problem in \(L^p\) \(L^p\)-dissipativity \(L^p\)-resolvent estimates Maximal domain Applications to rotating waves 

Notes

Acknowledgments

The author is greatly indebted to Giorgio Metafune, Alessandra Lunardi and Wolf-Jürgen Beyn for extensive discussions which helped in clarifying proofs. I also thank the anonymous referee for valuable suggestions which improved the first version of the paper. Supported by CRC 701 ‘Spectral Structures and Topological Methods in Mathematics’.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsBielefeld UniversityBielefeldGermany

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