Estimates of the derivatives for parabolic operators with unbounded coefficients
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- by Marcello Bertoldi and Luca Lorenzi PDF
- Trans. Amer. Math. Soc. 357 (2005), 2627-2664 Request permission
Abstract:
We consider a class of second-order uniformly elliptic operators $\mathcal {A}$ with unbounded coefficients in $\mathbb {R}^N$. Using a Bernstein approach we provide several uniform estimates for the semigroup $T(t)$ generated by the realization of the operator $\mathcal {A}$ in the space of all bounded and continuous or Hölder continuous functions in $\mathbb {R}^N$. As a consequence, we obtain optimal Schauder estimates for the solution to both the elliptic equation $\lambda u-\mathcal {A}u=f$ ($\lambda >0$) and the nonhomogeneous Dirichlet Cauchy problem $D_tu=\mathcal {A}u+g$. Then, we prove two different kinds of pointwise estimates of $T(t)$ that can be used to prove a Liouville-type theorem. Finally, we provide sharp estimates of the semigroup $T(t)$ in weighted $L^p$-spaces related to the invariant measure associated with the semigroup.References
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Additional Information
- Marcello Bertoldi
- Affiliation: Applied Mathematical Analysis, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands
- Email: bertoldi@fastmail.fm
- Luca Lorenzi
- Affiliation: Dipartimento di Matematica, Università di Parma, Via M. D’Azeglio 85/A, 43100 Parma, Italy
- MR Author ID: 649239
- Email: luca.lorenzi@unipr.it
- Received by editor(s): July 7, 2003
- Published electronically: March 1, 2005
- Additional Notes: This work was partially supported by the research project “Equazioni di evoluzione deterministiche e stocastiche" of the Ministero dell’Istruzione, dell’Università e della Ricerca (M.I.U.R.) and by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00281 “Evolution Equations".
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 357 (2005), 2627-2664
- MSC (2000): Primary 35B45; Secondary 35B65, 35K10, 47D06
- DOI: https://doi.org/10.1090/S0002-9947-05-03781-5
- MathSciNet review: 2139521