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Estimates of the derivatives for a class of parabolic degenerate operators with unbounded coefficients in $$\mathbb R^N$$. (English) Zbl 1107.35071
The paper deals with a class of perturbations of the degenerate Ornstein-Uhlenbeck operator in $$\mathbb R^N.$$ By virtue of a revised version of the Bernstein method, the author obtains various uniform estimates for the semigroup $$\{T(t)\}_{t\geq0}$$ associated with the realization of a degenerate elliptic operator in the space of the bounded and continuous functions in $$\mathbb R^N.$$

##### MSC:
 35K65 Degenerate parabolic equations 35B65 Smoothness and regularity of solutions to PDEs 47D06 One-parameter semigroups and linear evolution equations 35B45 A priori estimates in context of PDEs
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##### References:
 [1] S. Bernstein, Sur la généralisation du probléme de Dirichlet, I, Math. Ann. 62 (1906), 253-271. MR1511375 JFM37.0383.01 · JFM 37.0383.01 [2] M. Bertoldi and L. Lorenzi, Analytic methods for Markov semigroups, Preprint 401, Dipartimento di Matematica, Università di Parma, 2005. MR2313847 · Zbl 1065.35077 [3] M. Bertoldi and L. Lorenzi, Estimates of the derivatives for parabolic operators with unbounded coefficients, Trans. Amer. Math. Soc. (to appear). Zbl1065.35077 MR2139521 · Zbl 1065.35077 [4] S. Cerrai, Some results for second order elliptic operators having unbounded coefficients, Differential Integral Equations 11 (1998), 561-588. Zbl1131.35393 MR1666273 · Zbl 1131.35393 [5] G. Da Prato, Regularity results for some degenerate parabolic equations, Riv. Mat. Univ. Parma (6) 2* (1999), 245-257. Zbl0962.35110 MR1752802 · Zbl 0962.35110 [6] S. Fornaro, G. Metafune and E. Priola, Gradient estimates for Dirichlet parabolic problems in unbounded domains, J. Differential Equations 205 (2004), 329-353. Zbl1061.35022 MR2092861 · Zbl 1061.35022 [7] A. Friedman, “Partial Differential Equations of Parabolic Type”, Prentice Hall, Englewood Cliffs, N.J., 1964. Zbl0144.34903 MR181836 · Zbl 0144.34903 [8] R.Z. Has’minskii, “Stochastic Stability of Differential Equations”, Nauka 1969 (in Russian), English translation: Sijthoff and Noordhoff 1980. MR600653 [9] N.V. Krylov, “Introduction to the Theory of Diffusion Processes”, American Mathematical Society 142, (1992). Zbl0844.60050 MR1311478 · Zbl 0844.60050 [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. Zbl0174.15403 · Zbl 0174.15403 [11] G. Lieberman, “Second Order Parabolic Differential Equations”, World Scientific Publishing Co. Pte. Ltd, Singapore, New Jersey, London Hong Kong, 1996. Zbl0884.35001 MR1465184 · Zbl 0884.35001 [12] L. Lorenzi, Schauder estimates for a class of degenerate elliptic and parabolic problems with unbounded coefficients, Differential Integral Equations 18 (2005), 531-566. Zblpre05720819 MR2136978 · Zbl 1212.35255 [13] A. Lunardi, “Analytic Semigroups and Optimal Regularity in Parabolic Problems”, Birkhäuser, Basel, 1995. Zbl0816.35001 MR1329547 · Zbl 0816.35001 [14] A. Lunardi, Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in $$\mathbb{R}^N$$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24 (1997), 133-164. Zbl0887.35062 MR1475774 · Zbl 0887.35062 [15] A. Lunardi, Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $$\mathbb{R}^N$$, Studia Math. 128 (1998), 171-198. Zbl0899.35014 MR1490820 · Zbl 0899.35014 [16] M. Manfredini, The Dirichlet problem for a class of ultraparabolic equations, Adv. Differential Equations 2 (1997), 831-866. Zbl1023.35518 MR1751429 · Zbl 1023.35518 [17] M. Manfredini and A. Pascucci, A priori estimates for quasilinear degenerate parabolic equations, Proc. Amer. Math. Soc. 131 (2002), 1115-1120. Zbl1195.35173 MR1948102 · Zbl 1195.35173 [18] G. Metafune, D. Pallara and M. Wacker, Feller semigroups on $$\mathbb{R}^N$$, Semigroup Forum 65 (2002), 159-205. Zbl1014.35050 MR1911723 · Zbl 1014.35050 [19] A. Pascucci, Hölder regularity for a Kolmogorov equation, Trans. Amer. Math. Soc. 355 (2002), 901-924. Zbl1116.35330 MR1938738 · Zbl 1116.35330 [20] S. Polidoro, On a class of ultraparabolic operators of Kolmogorov-Fokker-Plank type, Matematiche (Catania) 49 (1994), 53-105 (1995). Zbl0845.35059 MR1386366 · Zbl 0845.35059 [21] E. Priola, The Cauchy problem for a class of Markov-type semigroups, Comm. Appl. Anal. 5 (2001), 49-75. Zbl1084.47517 MR1844671 · Zbl 1084.47517
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