American Institute of Mathematical Sciences

Advanced
  • Previous Article
    On the reducibility of analytic quasi-periodic systems with Liouvillean basic frequencies
  • CPAA Home
  • This Issue
  • Next Article
    Periodic solution and extinction in a periodic chemostat model with delay in microorganism growth
April  2022, 21(4): 1385-1416. doi: 10.3934/cpaa.2022023

Schauder type estimates for degenerate Kolmogorov equations with Dini continuous coefficients

Sergio Polidoro 1, , Annalaura Rebucci 1, and  Bianca Stroffolini 2,,

1. 

Università degli Studi di Modena e Reggio Emilia, Dipartimento di Scienze Fisiche, Informatiche e Matematiche, via Campi 213/b, 41125 Modena, Italy
2. 

Università di Napoli Federico II, Dipartimento di Ingegneria Elettrica e delle Tecnologie dell'Informazione, Via Claudio 25, 80125 Napoli, Italy

*Corresponding author

Received  September 2021 Published  April 2022 Early access  January 2022

Fund Project: This research was partially supported by the grant of Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)

We study the regularity properties of the second order linear operator in
$ {{\mathbb {R}}}^{N+1} $
:
$ \begin{equation*} \mathscr{L} u : = \sum\limits_{j,k = 1}^{m} a_{jk}\partial_{x_j x_k}^2 u + \sum\limits_{j,k = 1}^{N} b_{jk}x_k \partial_{x_j} u - \partial_t u, \end{equation*} $
where
$ A = \left( a_{jk} \right)_{j,k = 1, \dots, m}, B = \left( b_{jk} \right)_{j,k = 1, \dots, N} $
 are real valued matrices with constant coefficients, with
$ A $
 symmetric and strictly positive. We prove that, if the operator
$ {\mathscr{L}} $
 satisfies Hörmander's hypoellipticity condition, and
$ f $
 is a Dini continuous function, then the second order derivatives of the solution
$ u $
 to the equation
$ {\mathscr{L}} u = f $
 are Dini continuous functions as well. We also consider the case of Dini continuous coefficients
$ a_{jk} $
's. A key step in our proof is a Taylor formula for classical solutions to
$ {\mathscr{L}} u = f $
 that we establish under minimal regularity assumptions on
$ u $
.
Keywords: Degenerate Kolmogorov equations, regularity theory, classical solutions, Dini continuity, Taylor formula.
Mathematics Subject Classification: 35K70, 35K65, 35B65.
Citation: Sergio Polidoro, Annalaura Rebucci, Bianca Stroffolini. Schauder type estimates for degenerate Kolmogorov equations with Dini continuous coefficients. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1385-1416. doi: 10.3934/cpaa.2022023
References:
[1]

F. Anceschi and S. Polidoro, A survey on the classical theory for Kolmogorov equation, Matematiche (Catania), 75 (2020), 221-258.  doi: 10.4418/2020.75.1.11.

[2]

G. Arena, A. O. Caruso and A. Causa, Taylor formula on step two Carnot groups, Rev. Mat. Iberoam., 26 (2010), 239-259.  doi: 10.4171/RMI/600.

[3]

A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni, Stratified Lie groups and potential theory for their sub-Laplacians, in Springer Monographs in Mathematics, Springer, Berlin, 2007.

[4]

A. Bonfiglioli, Taylor formula for homogeneous groups and applications, Math. Z., 262 (2009), 255-279.  doi: 10.1007/s00209-008-0372-z.

[5]

C. Bucur and A. L. Karakhanyan, Potential theoretic approach to Schauder estimates for the fractional Laplacian, Proc. Amer. Math. Soc., 145 (2017), 637-651.  doi: 10.1090/proc/13227.

[6]

M. Di Francesco and S. Polidoro, Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov-type operators in non-divergence form, Adv. Differ. Equ., 11 (2006), 1261-1320. 

[7]

G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207.  doi: 10.1007/BF02386204.

[8]

G. B. Folland and E. M. Stein, Estimates for the $\bar \partial _{b}$ complex and analysis on the Heisenberg group, Commun. Pure Appl. Math., 27 (1974), 429-522.  doi: 10.1002/cpa.3160270403.

[9]

G. B. Folland and E. M. Stein, Hardy spaces on homogeneous groups, in Mathematical Notes, Princeton University Press, 1982.

[10]

La rs Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.  doi: 10.1007/BF02392081.

[11]

C. Imbert and L. Silvestre, Regularity for the Boltzmann equation conditional to macroscopic bounds, EMS Press, 7 (2020), 117-172.  doi: 10.4171/emss/37.

[12]

A. Kolmogoroff, Zufällige Bewegungen (zur Theorie der Brownschen Bewegung), Ann. Math., 35 (1934), 116-117.  doi: 10.2307/1968123.

[13]

E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators, Partial differential equations, II (Turin, 1993), 52 (1994), 29-63. 

[14]

L. Lorenzi, Schauder estimates for degenerate elliptic and parabolic problems with unbounded coefficients in ${\mathbb R}^N$, Differ. Integral Equ., 18 (2005), 531-566. 

[15]

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., 24 (1997), 133-164. 

[16]

M. Manfredini, The Dirichlet problem for a class of ultraparabolic equations, Adv. Differ. Equ., 2 (1997), 831-866. 

[17]

S. Menozzi, Parametrix techniques and martingale problems for some degenerate Kolmogorov equations, Electron. Commun. Probab., 16 (2011), 234-250.  doi: 10.1214/ECP.v16-1619.

[18]

A. Nagel, E. M. Stein and S. Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math., 155 (1985), 103-147.  doi: 10.1007/BF02392539.

[19]

S. Pagliarani, A. Pascucci and M. Pignotti, Intrinsic Taylor formula for Kolmogorov-type homogeneous groups, J. Math. Anal. Appl., 435 (2016), 1054-1087.  doi: 10.1016/j.jmaa.2015.10.080.

[20]

Stefano Pagliarani and Michele Pignotti, Intrinsic Taylor formula for non-homogeneous Kolmogorov-type Lie groups, arXiv: 1707.01422v2.

[21]

A. Pascucci, PDE and Martingale Methods in Option Pricing, Milano, Springer, 2011. doi: 10.1007/978-88-470-1781-8.

[22]

M. Pignotti, Averaged Stochastic Processes and Kolmogorov Operators, Ph.D thesis, Alma Mater Studiorum Università di Bologna, 2018.

[23]

E. Priola, Global Schauder estimates for a class of degenerate Kolmogorov equations, Stud. Math., 194 (2007), 117-153.  doi: 10.4064/sm194-2-2.

[24]

L. P. Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math., 137 (1976), 247-320.  doi: 10.1007/BF02392419.

[25]

X. J. Wang, Schauder estimates for elliptic and parabolic equations, Chinese Ann. Math. Ser. B, 27 (2006), 637-642.  doi: 10.1007/s11401-006-0142-3.

[26]

N. Wei, Y. Jiang and Y. Wu, Partial Schauder estimates for a sub-elliptic equation, Acta Math. Sci. Ser. B (Engl. Ed.), 36 (2016), 945-956.  doi: 10.1016/S0252-9602(16)30051-0.

show all references

References:
[1]

F. Anceschi and S. Polidoro, A survey on the classical theory for Kolmogorov equation, Matematiche (Catania), 75 (2020), 221-258.  doi: 10.4418/2020.75.1.11.

[2]

G. Arena, A. O. Caruso and A. Causa, Taylor formula on step two Carnot groups, Rev. Mat. Iberoam., 26 (2010), 239-259.  doi: 10.4171/RMI/600.

[3]

A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni, Stratified Lie groups and potential theory for their sub-Laplacians, in Springer Monographs in Mathematics, Springer, Berlin, 2007.

[4]

A. Bonfiglioli, Taylor formula for homogeneous groups and applications, Math. Z., 262 (2009), 255-279.  doi: 10.1007/s00209-008-0372-z.

[5]

C. Bucur and A. L. Karakhanyan, Potential theoretic approach to Schauder estimates for the fractional Laplacian, Proc. Amer. Math. Soc., 145 (2017), 637-651.  doi: 10.1090/proc/13227.

[6]

M. Di Francesco and S. Polidoro, Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov-type operators in non-divergence form, Adv. Differ. Equ., 11 (2006), 1261-1320. 

[7]

G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207.  doi: 10.1007/BF02386204.

[8]

G. B. Folland and E. M. Stein, Estimates for the $\bar \partial _{b}$ complex and analysis on the Heisenberg group, Commun. Pure Appl. Math., 27 (1974), 429-522.  doi: 10.1002/cpa.3160270403.

[9]

G. B. Folland and E. M. Stein, Hardy spaces on homogeneous groups, in Mathematical Notes, Princeton University Press, 1982.

[10]

La rs Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.  doi: 10.1007/BF02392081.

[11]

C. Imbert and L. Silvestre, Regularity for the Boltzmann equation conditional to macroscopic bounds, EMS Press, 7 (2020), 117-172.  doi: 10.4171/emss/37.

[12]

A. Kolmogoroff, Zufällige Bewegungen (zur Theorie der Brownschen Bewegung), Ann. Math., 35 (1934), 116-117.  doi: 10.2307/1968123.

[13]

E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators, Partial differential equations, II (Turin, 1993), 52 (1994), 29-63. 

[14]

L. Lorenzi, Schauder estimates for degenerate elliptic and parabolic problems with unbounded coefficients in ${\mathbb R}^N$, Differ. Integral Equ., 18 (2005), 531-566. 

[15]

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., 24 (1997), 133-164. 

[16]

M. Manfredini, The Dirichlet problem for a class of ultraparabolic equations, Adv. Differ. Equ., 2 (1997), 831-866. 

[17]

S. Menozzi, Parametrix techniques and martingale problems for some degenerate Kolmogorov equations, Electron. Commun. Probab., 16 (2011), 234-250.  doi: 10.1214/ECP.v16-1619.

[18]

A. Nagel, E. M. Stein and S. Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math., 155 (1985), 103-147.  doi: 10.1007/BF02392539.

[19]

S. Pagliarani, A. Pascucci and M. Pignotti, Intrinsic Taylor formula for Kolmogorov-type homogeneous groups, J. Math. Anal. Appl., 435 (2016), 1054-1087.  doi: 10.1016/j.jmaa.2015.10.080.

[20]

Stefano Pagliarani and Michele Pignotti, Intrinsic Taylor formula for non-homogeneous Kolmogorov-type Lie groups, arXiv: 1707.01422v2.

[21]

A. Pascucci, PDE and Martingale Methods in Option Pricing, Milano, Springer, 2011. doi: 10.1007/978-88-470-1781-8.

[22]

M. Pignotti, Averaged Stochastic Processes and Kolmogorov Operators, Ph.D thesis, Alma Mater Studiorum Università di Bologna, 2018.

[23]

E. Priola, Global Schauder estimates for a class of degenerate Kolmogorov equations, Stud. Math., 194 (2007), 117-153.  doi: 10.4064/sm194-2-2.

[24]

L. P. Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math., 137 (1976), 247-320.  doi: 10.1007/BF02392419.

[25]

X. J. Wang, Schauder estimates for elliptic and parabolic equations, Chinese Ann. Math. Ser. B, 27 (2006), 637-642.  doi: 10.1007/s11401-006-0142-3.

[26]

N. Wei, Y. Jiang and Y. Wu, Partial Schauder estimates for a sub-elliptic equation, Acta Math. Sci. Ser. B (Engl. Ed.), 36 (2016), 945-956.  doi: 10.1016/S0252-9602(16)30051-0.

[1]

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
[2]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515
[3]

Annamaria Canino, Elisa De Giorgio, Berardino Sciunzi. Second order regularity for degenerate nonlinear elliptic equations. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4231-4242. doi: 10.3934/dcds.2018184
[4]

Huilian Jia, Lihe Wang, Fengping Yao, Shulin Zhou. Regularity theory in Orlicz spaces for the poisson and heat equations. Communications on Pure and Applied Analysis, 2008, 7 (2) : 407-416. doi: 10.3934/cpaa.2008.7.407
[5]

Vladimir I. Bogachev, Stanislav V. Shaposhnikov, Alexander Yu. Veretennikov. Differentiability of solutions of stationary Fokker--Planck--Kolmogorov equations with respect to a parameter. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3519-3543. doi: 10.3934/dcds.2016.36.3519
[6]

Dung Le. Partial regularity of solutions to a class of strongly coupled degenerate parabolic systems. Conference Publications, 2005, 2005 (Special) : 576-586. doi: 10.3934/proc.2005.2005.576
[7]

David McCaffrey. A representational formula for variational solutions to Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1205-1215. doi: 10.3934/cpaa.2012.11.1205
[8]

Yanbo Hu, Tong Li. The regularity of a degenerate Goursat problem for the 2-D isothermal Euler equations. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3317-3336. doi: 10.3934/cpaa.2019149
[9]

Mounim El Ouardy, Youssef El Hadfi, Aziz Ifzarne. Existence and regularity results for a singular parabolic equations with degenerate coercivity. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 117-141. doi: 10.3934/dcdss.2021012
[10]

José Antonio Carrillo, Marco Di Francesco, Antonio Esposito, Simone Fagioli, Markus Schmidtchen. Measure solutions to a system of continuity equations driven by Newtonian nonlocal interactions. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 1191-1231. doi: 10.3934/dcds.2020075
[11]

Wenxiong Chen, Congming Li. Regularity of solutions for a system of integral equations. Communications on Pure and Applied Analysis, 2005, 4 (1) : 1-8. doi: 10.3934/cpaa.2005.4.1
[12]

Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310
[13]

Julien Dambrine, Morgan Pierre. Continuity with respect to the speed for optimal ship forms based on Michell's formula. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021049
[14]

Yaozhong Hu, Yanghui Liu, David Nualart. Taylor schemes for rough differential equations and fractional diffusions. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3115-3162. doi: 10.3934/dcdsb.2016090
[15]

Hongjie Dong, Xinghong Pan. On conormal derivative problem for parabolic equations with Dini mean oscillation coefficients. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4567-4592. doi: 10.3934/dcds.2021049
[16]

Bingyuan Huang, Shijin Ding, Huanyao Wen. Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1717-1752. doi: 10.3934/dcdss.2016072
[17]

Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085
[18]

P. Di Gironimo, L. D’Onofrio. On the regularity of minimizers to degenerate functionals. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1311-1318. doi: 10.3934/cpaa.2010.9.1311
[19]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2505-2518. doi: 10.3934/cpaa.2020272
[20]

Maria Colombo, Gianluca Crippa, Stefano Spirito. Logarithmic estimates for continuity equations. Networks and Heterogeneous Media, 2016, 11 (2) : 301-311. doi: 10.3934/nhm.2016.11.301

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (215)
  • HTML views (245)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy