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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Estimates of the derivatives for parabolic operators with unbounded coefficients


Authors: Marcello Bertoldi and Luca Lorenzi
Journal: Trans. Amer. Math. Soc. 357 (2005), 2627-2664
MSC (2000): Primary 35B45; Secondary 35B65, 35K10, 47D06
Published electronically: March 1, 2005
MathSciNet review: 2139521
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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.


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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
Email: luca.lorenzi@unipr.it

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03781-5
PII: S 0002-9947(05)03781-5
Keywords: Elliptic and parabolic operators with unbounded coefficients in ${\mathbb R}^N$, Markov semigroups, uniform and pointwise estimates, optimal Schauder estimates
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".
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.