Optimal Hölder regularity for nonautonomous Kolmogorov equations

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

Optimal Hölder regularity for nonautonomous Kolmogorov equations

Luca Lorenzi ^1,

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

Abstract
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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$u+g$, $u(0,\cdot)=f$.

Keywords: discontinuous coefficients, optimal Schauder estimates., nonautonomous elliptic and parabolic operators with unbounded coefficients.

Mathematics Subject Classification: Primary: 35B65; Secondary: 35K15, 35R0.

Citation: Luca Lorenzi. Optimal Hölder regularity for nonautonomous Kolmogorov equations. Discrete and 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.
[2]	M. Bertoldi and L. Lorenzi, "Analytical Methods for Markov Semigroups," Chapman and Hall/CRC Press, Boca Raton, FL, 2007.
[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.
[4]	A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall Inc., 1964.
[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.
[6]	N. V. Krylov and E. Priola, Elliptic and parabolic second-order PDEs with growing coefficients,, preprint, ().
[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).
[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.
[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.
[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.
[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.
[12]	A. Lunardi, An interpolation method to characterize domains of generators of semigroups, Semigroup Forum, 53 (1996), 321-329. doi: doi:10.1007/BF02574147.
[13]	A. Lunardi, Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $\R^n$, Studia Mathematica, 128 (1998), 171-198.
[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.
[15]	H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland, Amsterdam, 1978.

show all references

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.
[2]	M. Bertoldi and L. Lorenzi, "Analytical Methods for Markov Semigroups," Chapman and Hall/CRC Press, Boca Raton, FL, 2007.
[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.
[4]	A. Friedman, "Partial Differential Equations of Parabolic Type," Prentice-Hall Inc., 1964.
[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.
[6]	N. V. Krylov and E. Priola, Elliptic and parabolic second-order PDEs with growing coefficients,, preprint, ().
[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).
[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.
[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.
[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.
[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.
[12]	A. Lunardi, An interpolation method to characterize domains of generators of semigroups, Semigroup Forum, 53 (1996), 321-329. doi: doi:10.1007/BF02574147.
[13]	A. Lunardi, Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $\R^n$, Studia Mathematica, 128 (1998), 171-198.
[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.
[15]	H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland, Amsterdam, 1978.

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