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


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.


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



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’.


  1. 1.
    Alt, H.W.: Lineare Funktionalanalysis. Springer, Berlin (2006)zbMATHGoogle Scholar
  2. 2.
    Beyn, W.-J., Lorenz, J.: Nonlinear stability of rotating patterns. Dyn. Partial Differ. Equ. 5(4), 349–400 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Beyn, W.-J., Otten, D.: Spatial Decay of Rotating Waves in Reaction Diffusion Systems. Preprint., (2016) (submitted)
  4. 4.
    Cialdea, A.: Criteria for the \(L^p\)-dissipativity of partial differential operators. In: Cialdea, A., Ricci, P., Lanzara, F. (eds.) Analysis, Partial Differential Equations and Applications. Operator Theory: Advances and Applications, vol. 193, pp. 41–56. Birkhuser, Basel (2009)CrossRefGoogle Scholar
  5. 5.
    Cialdea, A., Maz’ya, V.: Criterion for the \(L^p\)-dissipativity of second order differential operators with complex coefficients. J. Math. Pures Appl. (9) 84(8), 1067–1100 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Da Prato, G., Lunardi, A.: On the Ornstein–Uhlenbeck operator in spaces of continuous functions. J. Funct. Anal. 131(1), 94–114 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, vol. 194. Springer, New York (2000). With contributions by Brendle, S., Campiti, M., Hahn, T., Metafune, G., Nickel, G., Pallara, D., Perazzoli, C., Rhandi, A., Romanelli, S., Schnaubelt, RzbMATHGoogle Scholar
  8. 8.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2010). [u.a.]zbMATHGoogle Scholar
  9. 9.
    Ladyzenskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasi-linear Equations of Parabolic Type. Translations of Mathematical Monographs. American Mathematical Society, Providence, RI (1968)Google Scholar
  10. 10.
    Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progress in Nonlinear Differential Equations and their Applications. Birkhäuser, Basel (1995)zbMATHGoogle Scholar
  11. 11.
    Metafune, G.: \(L^p\)-spectrum of Ornstein–Uhlenbeck operators. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 30(1), 97–124 (2001)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Metafune, G., Pallara, D., Vespri, V.: \(L^p\)-estimates for a class of elliptic operators with unbounded coefficients in \(\mathbf{R}^N\). Houston J. Math. 31(2), 605–620 (2005). (electronic)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Metafune, G., Pallara, D., Wacker, M.: Feller semigroups on \(R^N\). Semigr. Forum 65(2), 159–205 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Metafune, G., Prüss, J., Rhandi, A., Schnaubelt, R.: The domain of the Ornstein–Uhlenbeck operator on an \(L^p\)-space with invariant measure. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1(2), 471–485 (2002)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Mielke, A.: The Ginzburg–Landau equation in its role as a modulation equation. In: Fiedler, B. (ed.) Handbook of Dynamical Systems, pp. 759–834. North-Holland, Amsterdam (2002)Google Scholar
  16. 16.
    Otten, D.: Spatial decay and spectral properties of rotating waves in parabolic systems. PhD thesis, Bielefeld University (2014). Shaker Verlag, Aachen
  17. 17.
    Otten, D.: Exponentially weighted resolvent estimates for complex Ornstein–Uhlenbeck systems. J. Evol. Equ. 15(4), 753–799 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Otten, D.: A new \({L}^p\)-Antieigenvalue Condition for Ornstein–Uhlenbeck Operators. Preprint. (2015) (submitted)
  19. 19.
    Prüss, J., Rhandi, A., Schnaubelt, R.: The domain of elliptic operators on \(L^p(R^d)\) with unbounded drift coefficients. Houston J. Math. 32(2), 563–576 (2006). (electronic)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Rudin, W.: Functional Analysis. International Series in Pure and Applied Mathematics, 2nd edn. McGraw-Hill Inc., New York (1991)Google Scholar
  21. 21.
    Uhlenbeck, G.E., Ornstein, L.S.: On the theory of the brownian motion. Phys. Rev. 36, 823–841 (1930)CrossRefzbMATHGoogle Scholar
  22. 22.
    Zelik, S., Mielke, A.: Multi-pulse evolution and space–time chaos in dissipative systems. Mem. Am. Math. Soc. 198(925), vi+97 (2009)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Ziemer, W.P.: Weakly Differentiable Functions. Graduate Texts in Mathematics. Sobolev Spaces and Functions of Bounded Variation, vol. 120. Springer, New York (1989)zbMATHGoogle Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsBielefeld UniversityBielefeldGermany

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