# American Institute of Mathematical Sciences

February  2011, 4(1): 169-191. doi: 10.3934/dcdss.2011.4.169

## Optimal Hölder regularity for nonautonomous Kolmogorov equations

 1 Dipartimento di Matematica, Università degli Studi di Parma, Viale Parco Area delle Scienze 53/A, I-43124 Parma, Italy

Received  February 2009 Revised  October 2009 Published  October 2010

We consider a class of nonautonomous elliptic operators A with unbounded coefficients defined in $[0,T]\times\R^N$ and we prove optimal Schauder estimates for the solution to the parabolic Cauchy problem $D_tu=$A$u+g$, $u(0,\cdot)=f$.
Citation: Luca Lorenzi. Optimal Hölder regularity for nonautonomous Kolmogorov equations. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 169-191. doi: 10.3934/dcdss.2011.4.169
##### References:
 [1] M. Bertoldi and L. Lorenzi, Estimates of the derivatives for parabolic operators with unbounded coefficients, Trans. Amer. Math. Soc., 357 (2005), 2627-2664. doi: doi:10.1090/S0002-9947-05-03781-5.  Google Scholar [2] M. Bertoldi and L. Lorenzi, "Analytical Methods for Markov Semigroups," Chapman and Hall/CRC Press, Boca Raton, FL, 2007.  Google Scholar [3] S. Cerrai, Elliptic and parabolic equations in $\R^n$ with coefficients having polynomial growth, Comm. Partial Differential Equations, 21 (1996), 281-317. doi: doi:10.1080/03605309608821185.  Google Scholar [4] A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall Inc., 1964.  Google Scholar [5] M. Kunze, L. Lorenzi and A. Lunardi, Nonautonomous Kolmogorov parabolic equations with unbounded coefficients, Trans. Amer. Math. Soc., 362 (2010), 169-198. doi: doi:10.1090/S0002-9947-09-04738-2.  Google Scholar [6] N. V. Krylov and E. Priola, Elliptic and parabolic second-order PDEs with growing coefficients,, preprint, ().   Google Scholar [7] S. N. Kružkov, A. Castro and M. Lopes, Schauder type estimates and theorems on the existence of the solutions of fundamental problems for linear and nonlinear parabolic equations, Dokl. Akad. Nauk. SSSR, 220 (1975), 277-280 (in Russian); Soviet Math. Dokl., 16 (1975), 60-64 (in English).  Google Scholar [8] S. N. Kružkov, A. Castro and M. Lopes, Mayoraciones de Schauder y teorema de existencia de las soluciones del problema de Cauchy para ecuaciones parabolicas lineales y no lineales I, Cienc. Mat. (Havana), 1 (1980), 55-76. Google Scholar [9] S. N. Kružkov, A. Castro and M. Lopes, Mayoraciones de Schauder y teorema de existencia de las soluciones del problema de Cauchy para ecuaciones parabolicas lineales y no lineales II, Cienc. Mat. (Havana), 3 (1982), 37-56. Google Scholar [10] L. Lorenzi, Optimal Schauder estimates for parabolic problems with data measurable with respect to time, SIAM J. Math. Anal., 32 (2000), 588-615. doi: doi:10.1137/S0036141098342842.  Google Scholar [11] L. Lorenzi, On a class of elliptic operators with unbounded time and space dependent coefficients in $\mathbb R^N$, in "Functional Analysis & Evolution Equation: dedicated to Gunter Lumer," Birkhäuser, 2007.  Google Scholar [12] A. Lunardi, An interpolation method to characterize domains of generators of semigroups, Semigroup Forum, 53 (1996), 321-329. doi: doi:10.1007/BF02574147.  Google Scholar [13] A. Lunardi, Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $\R^n$, Studia Mathematica, 128 (1998), 171-198.  Google Scholar [14] A. I. Nazarov and N. N. Ural'tseva, Convex-monotone hulls and an estimate of the maximum of the solution of a parabolic equation, J. Soviet Math., 37 (1987), 851-859. doi: doi:10.1007/BF01387723.  Google Scholar [15] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland, Amsterdam, 1978.  Google Scholar

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##### References:
 [1] M. Bertoldi and L. Lorenzi, Estimates of the derivatives for parabolic operators with unbounded coefficients, Trans. Amer. Math. Soc., 357 (2005), 2627-2664. doi: doi:10.1090/S0002-9947-05-03781-5.  Google Scholar [2] M. Bertoldi and L. Lorenzi, "Analytical Methods for Markov Semigroups," Chapman and Hall/CRC Press, Boca Raton, FL, 2007.  Google Scholar [3] S. Cerrai, Elliptic and parabolic equations in $\R^n$ with coefficients having polynomial growth, Comm. Partial Differential Equations, 21 (1996), 281-317. doi: doi:10.1080/03605309608821185.  Google Scholar [4] A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall Inc., 1964.  Google Scholar [5] M. Kunze, L. Lorenzi and A. Lunardi, Nonautonomous Kolmogorov parabolic equations with unbounded coefficients, Trans. Amer. Math. Soc., 362 (2010), 169-198. doi: doi:10.1090/S0002-9947-09-04738-2.  Google Scholar [6] N. V. Krylov and E. Priola, Elliptic and parabolic second-order PDEs with growing coefficients,, preprint, ().   Google Scholar [7] S. N. Kružkov, A. Castro and M. Lopes, Schauder type estimates and theorems on the existence of the solutions of fundamental problems for linear and nonlinear parabolic equations, Dokl. Akad. Nauk. SSSR, 220 (1975), 277-280 (in Russian); Soviet Math. Dokl., 16 (1975), 60-64 (in English).  Google Scholar [8] S. N. Kružkov, A. Castro and M. Lopes, Mayoraciones de Schauder y teorema de existencia de las soluciones del problema de Cauchy para ecuaciones parabolicas lineales y no lineales I, Cienc. Mat. (Havana), 1 (1980), 55-76. Google Scholar [9] S. N. Kružkov, A. Castro and M. Lopes, Mayoraciones de Schauder y teorema de existencia de las soluciones del problema de Cauchy para ecuaciones parabolicas lineales y no lineales II, Cienc. Mat. (Havana), 3 (1982), 37-56. Google Scholar [10] L. Lorenzi, Optimal Schauder estimates for parabolic problems with data measurable with respect to time, SIAM J. Math. Anal., 32 (2000), 588-615. doi: doi:10.1137/S0036141098342842.  Google Scholar [11] L. Lorenzi, On a class of elliptic operators with unbounded time and space dependent coefficients in $\mathbb R^N$, in "Functional Analysis & Evolution Equation: dedicated to Gunter Lumer," Birkhäuser, 2007.  Google Scholar [12] A. Lunardi, An interpolation method to characterize domains of generators of semigroups, Semigroup Forum, 53 (1996), 321-329. doi: doi:10.1007/BF02574147.  Google Scholar [13] A. Lunardi, Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $\R^n$, Studia Mathematica, 128 (1998), 171-198.  Google Scholar [14] A. I. Nazarov and N. N. Ural'tseva, Convex-monotone hulls and an estimate of the maximum of the solution of a parabolic equation, J. Soviet Math., 37 (1987), 851-859. doi: doi:10.1007/BF01387723.  Google Scholar [15] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland, Amsterdam, 1978.  Google Scholar
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