Estimates of the derivatives for a class of parabolic degenerate operators with unbounded coefficients in ${ℝ}^N$

Lorenzi, Luca

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 2, pp. 255-293.

We consider a class of perturbations of the degenerate Ornstein-Uhlenbeck operator in ${ℝ}^N$. Using a revised version of Bernstein’s method we provide several uniform estimates for the semigroup ${\left\{T\left(t\right)\right\}}_{t\ge 0}$ associated with the realization of the operator $𝒜$ in the space of all the bounded and continuous functions in ${ℝ}^N$

[1] S. Bernstein, Sur la généralisation du probléme de Dirichlet, I, Math. Ann. 62 (1906), 253-271. | JFM | MR

[2] M. Bertoldi and L. Lorenzi, Analytic methods for Markov semigroups, Preprint 401, Dipartimento di Matematica, Università di Parma, 2005. | MR

[3] M. Bertoldi and L. Lorenzi, Estimates of the derivatives for parabolic operators with unbounded coefficients, Trans. Amer. Math. Soc. (to appear). | MR | Zbl

[4] S. Cerrai, Some results for second order elliptic operators having unbounded coefficients, Differential Integral Equations 11 (1998), 561-588. | MR | Zbl

[5] G. Da Prato, Regularity results for some degenerate parabolic equations, Riv. Mat. Univ. Parma (6) 2* (1999), 245-257. | MR | Zbl

[6] S. Fornaro, G. Metafune and E. Priola, Gradient estimates for Dirichlet parabolic problems in unbounded domains, J. Differential Equations 205 (2004), 329-353. | MR | Zbl

[7] A. Friedman, “Partial Differential Equations of Parabolic Type”, Prentice Hall, Englewood Cliffs, N.J., 1964. | MR | Zbl

[8] R.Z. Has'Minskii, “Stochastic Stability of Differential Equations”, Nauka 1969 (in Russian), English translation: Sijthoff and Noordhoff 1980. | MR

[9] N.V. Krylov, “Introduction to the Theory of Diffusion Processes”, American Mathematical Society 142, (1992). | MR | Zbl

[10] O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural'Ceva, “Linear and Quasilinear Equations of Parabolic Type”, Nauka, English transl.: American Mathematical Society, Providence, 1968. | Zbl

[11] G. Lieberman, “Second Order Parabolic Differential Equations”, World Scientific Publishing Co. Pte. Ltd, Singapore, New Jersey, London Hong Kong, 1996. | MR | Zbl

[12] L. Lorenzi, Schauder estimates for a class of degenerate elliptic and parabolic problems with unbounded coefficients, Differential Integral Equations 18 (2005), 531-566. | MR

[13] A. Lunardi, “Analytic Semigroups and Optimal Regularity in Parabolic Problems”, Birkhäuser, Basel, 1995. | MR | Zbl

[14] A. Lunardi, Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in ${ℝ}^N$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24 (1997), 133-164. | Numdam | MR | Zbl

[15] A. Lunardi, Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in ${ℝ}^N$, Studia Math. 128 (1998), 171-198. | MR | Zbl

[16] M. Manfredini, The Dirichlet problem for a class of ultraparabolic equations, Adv. Differential Equations 2 (1997), 831-866. | MR | Zbl

[17] M. Manfredini and A. Pascucci, A priori estimates for quasilinear degenerate parabolic equations, Proc. Amer. Math. Soc. 131 (2002), 1115-1120. | MR | Zbl

[18] G. Metafune, D. Pallara and M. Wacker, Feller semigroups on ${ℝ}^N$, Semigroup Forum 65 (2002), 159-205. | MR | Zbl

[19] A. Pascucci, Hölder regularity for a Kolmogorov equation, Trans. Amer. Math. Soc. 355 (2002), 901-924. | MR | Zbl

[20] S. Polidoro, On a class of ultraparabolic operators of Kolmogorov-Fokker-Plank type, Matematiche (Catania) 49 (1994), 53-105 (1995). | MR | Zbl

[21] E. Priola, The Cauchy problem for a class of Markov-type semigroups, Comm. Appl. Anal. 5 (2001), 49-75. | MR | Zbl