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December  2020, 13(12): 3491-3494. doi: 10.3934/dcdss.2020112

A Zaremba-type criterion for hypoelliptic degenerate Ornstein–Uhlenbeck operators

Alessia E. Kogoj

Dipartimento di Scienze Pure e Applicate (DiSPeA), Università degli Studi di Urbino Carlo Bo, Piazza della Repubblica, 13 - 61029 Urbino (PU), Italy

Dedicated to Gisèle Ruiz Goldstein on the occasion of her 60th birthday

Received  January 2019 Revised  March 2019 Published  October 2019

We prove a cone-type criterion for a boundary point to be regular for the Dirichlet problem related to (possibly) degenerate Ornstein–Uhlenbeck operators in $ \mathbb{R}^N $. Our result extends the classical Zaremba cone criterion for the Laplace operator.

Keywords: Ornstein–Uhlenbeck operators, Dirichlet problem, Perron–Wiener solution, boundary regularity, exterior cone criterion, hypoelliptic operators, axiomatic potential theory.
Mathematics Subject Classification: Primary: 35H10, 35J70, 35D99, 35J25; Secondary: 31D05.
Citation: Alessia E. Kogoj. A Zaremba-type criterion for hypoelliptic degenerate Ornstein–Uhlenbeck operators. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3491-3494. doi: 10.3934/dcdss.2020112
References:
[1]

K. Beauchard and K. Pravda-Starov, Null-controllability of non-autonomous Ornstein–Uhlenbeck equations, J. Math. Anal. Appl., 456 (2017), 496-524.  doi: 10.1016/j.jmaa.2017.07.014.  Google Scholar

[2]

J.-M. Bony, Principe du maximum, inégalité de Harnack et unicité du problèeme de Cauchy pour les opérateurs elliptiques dégénérés, (French)Ann. Inst. Fourier (Grenoble), 19 (1969), 277-304.  doi: 10.5802/aif.319.  Google Scholar

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M. Bramanti, G. Cupini, E. Lanconelli and E. Priola, Global $L^p$ estimates for degenerate Ornstein-Uhlenbeck operators, Math. Z., 266 (2010), 789-816.  doi: 10.1007/s00209-009-0599-3.  Google Scholar

[4]

M. Bramanti, G. Cupini, E. Lanconelli and E. Priola, Global $L^p$ estimates for degenerate Ornstein-Uhlenbeck operators with variable coefficients, Math. Nachr., 286 (2013), 1087-1101.  doi: 10.1002/mana.201200189.  Google Scholar

[5]

C. Cinti and E. Lanconelli, Riesz and Poisson-Jensen representation formulas for a class of ultraparabolic operators on Lie groups, Potential Anal., 30 (2009), 179-200.  doi: 10.1007/s11118-008-9112-6.  Google Scholar

[6]

B. Farkas and A. Lunardi, Maximal regularity for Kolmogorov operators in $L^2$ spaces with respect to invariant measures, J. Math. Pures Appl. (9), 86 (2006), 310–321. doi: 10.1016/j.matpur.2006.06.002.  Google Scholar

[7]

S. Fornaro and A. Rhandi, On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$-spaces, Discrete Contin. Dyn. Syst. Ser. A, 33 (2013), 5049-5058.  doi: 10.3934/dcds.2013.33.5049.  Google Scholar

[8]

G. R. Goldstein, J. A. Goldstein and A. Rhandi, Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential., Appl. Anal., 91 (2012), 2057-2071.  doi: 10.1080/00036811.2011.587809.  Google Scholar

[9]

A. E. Kogoj, On the Dirichlet problem for hypoelliptic evolution equations: Perron–Wiener solution and a cone-type criterion, J. Differential Equations, 262 (2017), 1524-1539.  doi: 10.1016/j.jde.2016.10.018.  Google Scholar

[10]

A. E. Kogoj and S. Polidoro, Harnack inequality for hypoelliptic second order partial differential operators, Potential Anal., 45 (2016), 545-555.  doi: 10.1007/s11118-016-9557-y.  Google Scholar

[11]

E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators, Partial differential equations, Ⅱ (Turin, 1993), Rend. Sem. Mat. Univ. Politec. Torino, 52 (1994), 29-63.   Google Scholar

[12]

P. Negrini, Punti regolari per aperti cilindrici in uno spazio $\beta $-armonico, (Italian) Boll. Un. Mat. Ital. B (6), 2 (1983), 537–547.  Google Scholar

[13]

S. Zaremba, Sur le Principe de Dirichlet, (French) Acta Math., 34 (1911), 293-316.  doi: 10.1007/BF02393130.  Google Scholar

show all references

References:
[1]

K. Beauchard and K. Pravda-Starov, Null-controllability of non-autonomous Ornstein–Uhlenbeck equations, J. Math. Anal. Appl., 456 (2017), 496-524.  doi: 10.1016/j.jmaa.2017.07.014.  Google Scholar

[2]

J.-M. Bony, Principe du maximum, inégalité de Harnack et unicité du problèeme de Cauchy pour les opérateurs elliptiques dégénérés, (French)Ann. Inst. Fourier (Grenoble), 19 (1969), 277-304.  doi: 10.5802/aif.319.  Google Scholar

[3]

M. Bramanti, G. Cupini, E. Lanconelli and E. Priola, Global $L^p$ estimates for degenerate Ornstein-Uhlenbeck operators, Math. Z., 266 (2010), 789-816.  doi: 10.1007/s00209-009-0599-3.  Google Scholar

[4]

M. Bramanti, G. Cupini, E. Lanconelli and E. Priola, Global $L^p$ estimates for degenerate Ornstein-Uhlenbeck operators with variable coefficients, Math. Nachr., 286 (2013), 1087-1101.  doi: 10.1002/mana.201200189.  Google Scholar

[5]

C. Cinti and E. Lanconelli, Riesz and Poisson-Jensen representation formulas for a class of ultraparabolic operators on Lie groups, Potential Anal., 30 (2009), 179-200.  doi: 10.1007/s11118-008-9112-6.  Google Scholar

[6]

B. Farkas and A. Lunardi, Maximal regularity for Kolmogorov operators in $L^2$ spaces with respect to invariant measures, J. Math. Pures Appl. (9), 86 (2006), 310–321. doi: 10.1016/j.matpur.2006.06.002.  Google Scholar

[7]

S. Fornaro and A. Rhandi, On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$-spaces, Discrete Contin. Dyn. Syst. Ser. A, 33 (2013), 5049-5058.  doi: 10.3934/dcds.2013.33.5049.  Google Scholar

[8]

G. R. Goldstein, J. A. Goldstein and A. Rhandi, Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential., Appl. Anal., 91 (2012), 2057-2071.  doi: 10.1080/00036811.2011.587809.  Google Scholar

[9]

A. E. Kogoj, On the Dirichlet problem for hypoelliptic evolution equations: Perron–Wiener solution and a cone-type criterion, J. Differential Equations, 262 (2017), 1524-1539.  doi: 10.1016/j.jde.2016.10.018.  Google Scholar

[10]

A. E. Kogoj and S. Polidoro, Harnack inequality for hypoelliptic second order partial differential operators, Potential Anal., 45 (2016), 545-555.  doi: 10.1007/s11118-016-9557-y.  Google Scholar

[11]

E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators, Partial differential equations, Ⅱ (Turin, 1993), Rend. Sem. Mat. Univ. Politec. Torino, 52 (1994), 29-63.   Google Scholar

[12]

P. Negrini, Punti regolari per aperti cilindrici in uno spazio $\beta $-armonico, (Italian) Boll. Un. Mat. Ital. B (6), 2 (1983), 537–547.  Google Scholar

[13]

S. Zaremba, Sur le Principe de Dirichlet, (French) Acta Math., 34 (1911), 293-316.  doi: 10.1007/BF02393130.  Google Scholar

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