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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. D. Bakry, Transformations de Riesz pour les semi-groupes symétriques. II. Étude sous la condition Γ₂≥0, Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 145–174 (French). MR 889473, 10.1007/BFb0075844
  • 2. D. Bakry and M. Ledoux, Lévy-Gromov’s isoperimetric inequality for an infinite-dimensional diffusion generator, Invent. Math. 123 (1996), no. 2, 259–281. MR 1374200, 10.1007/s002220050026
  • 3. M. Bertoldi, Analytic methods for Markov semigroups, Ph.D. thesis, Università di Trento, 2002.
  • 4. M. Bertoldi and S. Fornaro, Gradient estimates in parabolic problems with unbounded coefficients, Studia Math. 165 (2004), no. 3, 221–254. MR 2109509, 10.4064/sm165-3-3
  • 5. Sandra Cerrai, Second order PDE’s in finite and infinite dimension, Lecture Notes in Mathematics, vol. 1762, Springer-Verlag, Berlin, 2001. A probabilistic approach. MR 1840644
  • 6. Giuseppe Da Prato, Regularity results for some degenerate parabolic equation, Riv. Mat. Univ. Parma (6) 2* (1999), 245–257 (2000). MR 1752802
  • 7. Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
  • 8. R. Z. Has′minskiĭ, Stochastic stability of differential equations, Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, vol. 7, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980. Translated from the Russian by D. Louvish. MR 600653
  • 9. N. V. Krylov, Introduction to the theory of diffusion processes, Translations of Mathematical Monographs, vol. 142, American Mathematical Society, Providence, RI, 1995. Translated from the Russian manuscript by Valim Khidekel and Gennady Pasechnik. MR 1311478
  • 10. O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Linear and quasilinear equations of parabolic type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian). MR 0241822
  • 11. Alessandra Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications, 16, Birkhäuser Verlag, Basel, 1995. MR 1329547
  • 12. Alessandra Lunardi, Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in 𝐑ⁿ, Studia Math. 128 (1998), no. 2, 171–198. MR 1490820
  • 13. G. Metafune, D. Pallara, and M. Wacker, Feller semigroups on 𝐑^{𝐍}, Semigroup Forum 65 (2002), no. 2, 159–205. MR 1911723, 10.1007/s002330010129
  • 14. G. Metafune and E. Priola, Some classes of non-analytic Markov semigroups, J. Math. Anal. Appl. 294 (2004), no. 2, 596–613. MR 2061345, 10.1016/j.jmaa.2004.02.037
  • 15. Giorgio Metafune, Jan Prüss, Abdelaziz Rhandi, and Roland Schnaubelt, The domain of the Ornstein-Uhlenbeck operator on an 𝐿^{𝑝}-space with invariant measure, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002), no. 2, 471–485. MR 1991148
  • 16. G. Metafune, J. Prüss, A. Rhandi, R. Schnaubelt, $L^p$ regularity for elliptic operators with unbounded coefficients, report 21 Institute of Analysis Martin Luther, Universitaet Halle Wittenberg FB Mathematik und Informatik (2002).
  • 17. Enrico Priola, The Cauchy problem for a class of Markov-type semigroups, Commun. Appl. Anal. 5 (2001), no. 1, 49–75. MR 1844671
  • 18. Enrico Priola and Jerzy Zabczyk, Liouville theorems for non-local operators, J. Funct. Anal. 216 (2004), no. 2, 455–490. MR 2095690, 10.1016/j.jfa.2004.04.001
  • 19. Feng-Yu Wang, On estimation of the logarithmic Sobolev constant and gradient estimates of heat semigroups, Probab. Theory Related Fields 108 (1997), no. 1, 87–101. MR 1452551, 10.1007/s004400050102

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35B45, 35B65, 35K10, 47D06

Retrieve articles in all journals with MSC (2000): 35B45, 35B65, 35K10, 47D06

Additional Information

Marcello Bertoldi
Affiliation: Applied Mathematical Analysis, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands

Luca Lorenzi
Affiliation: Dipartimento di Matematica, Università di Parma, Via M. D’Azeglio 85/A, 43100 Parma, Italy

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.